# Polynomials, Expressions, and Formulas

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Polynomials, Expressions, and Formulas
Or, How I Learned to Stop Worrying and Love the Variable

Overview We begin this module with a study of numbers and the basic operations of addition, subtraction, multiplication, and division. The algebraic properties of these operations are introduced along with the conventions for the order in which the operations are performed in mathematical expressions. Lessons are included on numerical and variable expressions, as well as on writing, evaluating, and simplifying these expressions. These concepts are extended to polynomials, with lessons on simplifying, adding, subtracting, multiplying, dividing, and factoring polynomials. There are also lessons on more general formulas, including evaluating and rewriting formulas. On the following Topics slide there is a link to a table of contents for each topic. Clicking on a link on a Contents slide will take you to the first lesson slide for that topic. In many cases, there will be more than one slide on that topic. Work through as many as you need to master the topic, and then go on to the next topic. Clicking on the heading for a slide will take you back to the Contents slide for that topic; clicking on Return to Topics List on the Contents slide will take you back to the main Topics page.

Topics Sets of Real Numbers Properties of Operations
Order of Operations Numerical Expressions Variables and Expressions Writing Variable Expressions Evaluating Variable Expressions Simplifying Variable Expressions Polynomial Expressions Polynomial Multiplication Polynomial Division Factoring Polynomials Formulas & Polynomials in Two (or more) Variables Polynomial Functions

Sets of Real Numbers CONTENTS:
Description and examples of subsets of the set of real numbers, including the rational numbers and irrational numbers Subsets of the Real Numbers Recognizing Irrational and Rational Numbers Interactive Practice Problems: Recognizing Rational and Irrational Numbers Return to Topics List

Sets of Real Numbers Link: Number Sets
Khan Academy video giving definitions of the various subsets of the real numbers Subsets include natural numbers, whole numbers, integers, rational numbers, and irrational numbers, with examples of numbers in each subset Time 5:17

Sets of Real Numbers Link: Real Numbers
Mathispower4u video on sets of real numbers Describes various sets of real numbers, size comparison of real numbers, and absolute values Time 9:07

Sets of Real Numbers Link: Recognizing Irrational Numbers Khan Academy video explaining the difference between rational and irrational numbers Time 2:18

Link: Recognizing Rational and Irrational Numbers
Sets of Real Numbers Link: Recognizing Rational and Irrational Numbers Khan Academy interactive practice problems on recognizing rational and irrational numbers A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Properties of Operations
CONTENTS: Properties of Operations Descriptions and examples of properties of operations on numbers, including commutative, associative, and distributive laws; identities and inverses for multiplication and addition Terminology of Properties of Numbers Commutative Property of Addition Commutative Property of Multiplication Associative Law of Addition Associative Law of Multiplication Identity Properties of 0 and 1 Inverse Properties of Addition and Multiplication The Distributive Property Interactive Practice Problems: Distributive Property Return to Topics List

Properties of Operations Link: Properties of Numbers 1 Khan Academy video examples recognizing the associative and commutative properties of arithmetic Time 2:03

Properties of Operations Link: Number Properties Terminology 1 Khan Academy video examples recognizing the distributive, associative, and commutative properties of arithmetic Time 3:05

Properties of Operations Link: Properties of Numbers 2 Khan Academy video examples recognizing the properties of arithmetic Time 1:52

Properties of Operations Link: Commutative Law of Addition Khan Academy video example applying the commutative and associative laws of addition to rewrite a number expression Time 1:40

Properties of Operations Link: Commutative Property for Addition Khan Academy video example applying the commutative and associative laws of addition to rewrite a number expression Time 3:23

Properties of Operations Link: Commutative Property of Multiplication Khan Academy video example applying the commutative law of multiplication to rewrite a number expression Time 1:48

Properties of Operations Link: Associative Law of Addition Khan Academy video example applying the associative law of addition to rewrite a number expression Time 2:10

Properties of Operations Link: Associative Law of Multiplication Khan Academy video example applying the associative law of multiplication to rewrite a number expression Time 2:15

Properties of Operations Link: Identity Property of 0 Khan Academy video examples demonstrating the additive identity property of 0 Uses the method of the number line and absolute values when adding a zero in a numerical expression with more than one operation Time 4:46

Properties of Operations Link: Identity Property of 1 Khan Academy video examples demonstrating the multiplicative identity property of 1, explaining that you always get the original term number as the answer Time 1:24

Link: Identity Property of 1 (Second Example)
Properties of Operations Link: Identity Property of 1 (Second Example) Khan Academy video with more examples demonstrating the multiplicative identity property of 1 Time 1:55

Properties of Operations Link: Inverse Property of Addition Khan Academy video examples of using the additive inverse to move from a given number back to zero The number line method is used to visualize the impact of adding the opposite of the given number Time 2:59

Properties of Operations Link: Inverse Property of Multiplication Khan Academy video examples demonstrating the multiplicative inverse property Uses the operation of division and builds to multiplication by the reciprocal Time 3:16

Properties of Operations Link: The Distributive Property Khan Academy video example demonstrating the distributive property of multiplication over addition, with an explanation of why the distributive property works Time 4:56

Properties of Operations Link: The Distributive Property 2 Khan Academy video example demonstrating the distributive property of multiplication over subtraction Time 2:11

Link: The Distributive Property Example 1
Properties of Operations Link: The Distributive Property Example 1 Khan Academy video examples demonstrating the distributive property in several algebraic expressions Examples require combining like terms and distributing fractions Time 5:39

Properties of Operations Link: Distributive Property Exercise Khan Academy video examples demonstrating the distributive property of multiplication over addition in numerical expressions One problem reviews factoring the greatest common divisor and writing a distributive expression Time 3:03

Properties of Operations Link: The Distributive Property Math Planet text and video examples demonstrating the distributive property in several situations Time 1:41

Properties of Operations Link: Distributive Property Khan Academy interactive practice problems on the distributive property A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Order of Operations CONTENTS:
Descriptions and examples of the conventions on the order of operations in numerical and variable expressions Introduction to Order of Operations Order of Operations Examples Order of Operations With and Without Parentheses Interactive Practice Problems: Order of Operations Interactive Practice Problems: Expressions With Parentheses Return to Topics List

Link: Introduction to Order of Operations
Khan Academy video introducing the order of operations on numbers Examples show the impact of the agreed upon order for simplifying numerical expressions Examples also explain operations at the same level and the “left to right” rule Time 9:40

Mathispower4u video introducing the order of operations on numbers Includes several examples applying the order of operations to simplify expressions Time 6:31

Khan Academy video example applying the order of operations on numbers to simplify a numerical expression Time 4:27

Link: Order of Operations with Exponents
Khan Academy video example applying the order of operations on numbers Time 4:36

Link: More Complicated Order of Operations Example
Khan Academy video advanced example applying the order of operations to a numerical expression involving parentheses Time 5:20

Khan Academy video advanced examples applying the order of operations on numbers Time 14:23

Link: Evaluating an Expression With and Without Parentheses
Order of Operations Link: Evaluating an Expression With and Without Parentheses Khan Academy video example demonstrating the effect of parentheses on the order of operations of numbers Time 2:06

Khan Academy interactive practice problems on order of operations of numbers A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Link: Order of Operations with Negative Numbers
Khan Academy interactive practice problems on order of operations of numbers A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Order of Operations Link: Expressions With Parentheses Khan Academy interactive practice problems on evaluating numerical expressions containing parentheses A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Numerical Expressions
CONTENTS: Numerical Expressions Examples and interactive problems on writing numerical expressions from verbal phrases Constructing Numerical Expressions Example Interactive Practice Problems: Writing Numerical Expressions Return to Topics List

Khan Academy Video example constructing a numerical expression from a verbal phrase Time 1:22

Link: Writing Numerical Expressions with Exponents
Khan Academy interactive practice problems on writing numerical expressions from verbal phrases A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Link: Writing Numerical Expressions with Exponents Word Problems
Khan Academy interactive practice problems on writing numerical expressions with exponents from word problems A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Variables and Expressions
CONTENTS: Variables and Expressions Description and examples of variables and expressions What is a Variable? Introduction to Variables and Expressions Notations for Multiplication With Variables Return to Topics List

Variables and Expressions Link: What is a Variable? Khan Academy video example introducing the concept of a variable Time 3:18

Link: Introduction to Variables and Variable Expressions
Variables and Expressions Link: Introduction to Variables and Variable Expressions Mathispower4u video introducing and explaining the concepts of variables and variable expressions Also explains the difference between an expression and an equation Includes examples of writing variable expressions from verbal phrases evaluating a variable expression at a number Time 7:55

Link: Why Aren't We Using the Multiplication Sign?
Variables and Expressions Link: Why Aren't We Using the Multiplication Sign? Khan Academy video examples demonstrating the use of variables and various notation for multiplication with variables Time 4:58

Writing Variable Expressions
CONTENTS: Writing Variable Expressions Translating verbal phrases into mathematical expressions Constructing Algebraic Expressions from Verbal Phrases Writing an Equation from a Word Problem Interactive Practice Problems: Writing Simple Variable Expressions Interactive Practice Problems: Writing Compound Variable Expressions Interactive Practice Problems: Writing Expressions from Word Problems Return to Topics List

Writing Variable Expressions
Link: How Do You Turn a Simple Verbal Phrase Into an Algebraic Expression? Virtual Nerd video introduction to translating verbal phrases into algebraic expressions Examples are given of very basic phrases that require each of the operations of addition, subtraction, multiplication, and division Translates words such as sum, total, difference, product, quotient, and ratio Time 3:36

Writing Variable Expressions
Link: How Do You Turn a Longer Verbal Phrase Into an Algebraic Expression? Virtual Nerd video example of translating a verbal phrase into an equation The example given is a compound phrase involving addition, multiplication, and fractions This example is also used in the video How Do You Write an Equation from a Word Problem? Time 5:10

Link: Ex 1: Writing Basic Algebraic Expressions
Writing Variable Expressions Link: Ex 1: Writing Basic Algebraic Expressions Mathispower4u video introduction to translating verbal phrases into algebraic expressions Examples are given of very basic phrases that require each of the operations of addition and subtraction Time 1:59

Link: Ex 2: Writing Basic Algebraic Expressions
Writing Variable Expressions Link: Ex 2: Writing Basic Algebraic Expressions Mathispower4u video example of translating a verbal phrase into an algebraic expression Example expresses the remaining part of a total distance in terms of the total and a given part of the total Time 2:35

Writing Variable Expressions Link: Writing Expressions 1 Khan Academy video examples writing variable expressions from verbal phrases Vocabulary reviewed in the examples includes quantity, sum, and product translated into mathematical expressions Time 1:38

Writing Variable Expressions Link: Writing Expressions 2 Khan Academy video examples writing variable expressions from verbal phrases Time 2:45

Link: Writing Expressions 2 Exercise Example
Writing Variable Expressions Link: Writing Expressions 2 Exercise Example Khan Academy video multiple choice example writing a variable expression from a verbal phrase Explains why some selections are incorrect and one is correct Time 2:23

Link: Writing Expressions 3 Exercise Example 1
Writing Variable Expressions Link: Writing Expressions 3 Exercise Example 1 Khan Academy video example writing an expression in two variables from a complex word problem Time 2:24

Link: Writing Expressions 3 Exercise Example 2
Writing Variable Expressions Link: Writing Expressions 3 Exercise Example 2 Khan Academy video example writing an expression in one variable from a verbal phrase Example involves a word problem about volumes of cubes Time 1:20

Link: Writing Expressions 3 Exercise Example 3
Writing Variable Expressions Link: Writing Expressions 3 Exercise Example 3 Khan Academy video example writing an expression in several variables from a verbal phrase Time 0:43

Link: How Do You Write an Equation from a Word Problem?
Writing Variable Expressions Link: How Do You Write an Equation from a Word Problem? Virtual Nerd video example of translating the phrases in a “word problem” into variables and an equation See also the video How Do You Turn a Longer Verbal Phrase Into an Algebraic Expression? Time 4:00

Link: Connect English with Mathematics and Graphing Lines (Part 1)
Writing Variable Expressions Link: Connect English with Mathematics and Graphing Lines (Part 1) AlRichards314 video introduction to translating verbal phrases into algebraic expressions Examples are given of translating very basic phrases including phrases such as “increased by” and “decreased by” Also includes translation of a word problem into a system of two linear equations Time 5:19

Writing Variable Expressions Link: Writing Expressions Khan Academy interactive practice problems on writing simple variable expressions from verbal phrases A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Writing Variable Expressions Link: Writing Expressions 2 Khan Academy interactive practice problems on writing compound variable expressions from verbal phrases A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Link: Writing Expressions with Variables Word Problems
Writing Variable Expressions Link: Writing Expressions with Variables Word Problems Khan Academy interactive practice problems on writing variable expressions from word problems A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Evaluating Variable Expressions
CONTENTS: Evaluating Variable Expressions Description and examples of evaluating variable expressions by substituting specific numbers for the variables Evaluating Algebraic Expressions Evaluating Expressions With Two Variables Evaluating Expressions Where Individual Variable Values are Unknown Evaluating Expressions in Word Problems Interactive Practice Problems: Evaluating Expressions in One Variable Interactive Practice Problems: Evaluating Expressions in Two Variables Interactive Practice Problems: Evaluating Expressions With Variables in Word Problems Return to Topics List

Evaluating Variable Expressions Link: Evaluating Algebraic Expressions Mathispower4u video examples introducing evaluating various types of algebraic expressions at specific numbers Defines variable expressions and reviews order of operations Steps use substitution, order of operations, exponents, and fractions Time 7:52

Link: Evaluating Expressions With One Variable
Evaluating Variable Expressions Link: Evaluating Expressions With One Variable Khan Academy video of very basic examples of evaluating algebraic expressions at specific numbers Time 3:21

Evaluating Variable Expressions Link: Example: Evaluating an Expression Khan Academy video example of evaluating an algebraic expression at specific numbers Time 2:00

Evaluating Variable Expressions Link: Expressions With Two Variables Khan Academy video examples introducing evaluating expressions in two variables Time 1:58

Link: Example: Evaluating Expressions With 2 Variables
Evaluating Variable Expressions Link: Example: Evaluating Expressions With 2 Variables Khan Academy video example of evaluating an algebraic expression with two variables at specific numbers Time 1:46

Link: Ex: Evaluate a Variable Expression with Exponents and Fractions
Evaluating Variable Expressions Link: Ex: Evaluate a Variable Expression with Exponents and Fractions Mathispower4u video example of substituting values into an expression with exponents and fractions Reviews order of operations 2:33

Link: Examples of Evaluating Variable Expressions
Khan Academy video examples of evaluating algebraic expressions of many types Time 12:26

Evaluating Variable Expressions
Link: Evaluating Expressions Where Individual Variable Values are Unknown Khan Academy video examples of evaluating variable expressions where relationships between the variables are known but individual variable values are not 4:39

Link: Evaluating Expressions with Unknown Variables 2
Evaluating Variable Expressions Link: Evaluating Expressions with Unknown Variables 2 Khan Academy video examples of evaluating variable expressions where relationships between the variables are known but individual variable values are not 3:00

Link: Example Evaluating Expressions in Word Problems
Evaluating Variable Expressions Link: Example Evaluating Expressions in Word Problems Khan Academy video example of evaluating an algebraic expression in a word problem Time 3:24

Link: Evaluating Expressions in One Variable
Evaluating Variable Expressions Link: Evaluating Expressions in One Variable Khan Academy interactive practice problems on evaluating algebraic expressions in one variable A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Link: Evaluating Expressions in Two Variables
Evaluating Variable Expressions Link: Evaluating Expressions in Two Variables Khan Academy interactive practice problems on evaluating algebraic expressions in two variables A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Link: Evaluating Expressions With Variables in Word Problems
Evaluating Variable Expressions Link: Evaluating Expressions With Variables in Word Problems Khan Academy interactive practice problems on evaluating algebraic expressions in word problems A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Simplifying Variable Expressions
CONTENTS: Simplifying Variable Expressions Description and examples of using properties of operations and combining like terms to simplify variable expressions Combining Like Terms Equivalent Forms of Expressions Interactive Practice Problems: Equivalent Forms of Expressions Return to Topics List

Simplifying Variable Expressions Link: Combining Like Terms Khan Academy video explaining like terms and how to combine like terms 4:32

Simplifying Variable Expressions Link: Adding Like Rational Terms Khan Academy video example of combining two variable expressions with fractions 1:14

Simplifying Variable Expressions Link: Combining Like Terms 1 Khan Academy video example of combining like terms to simplify an algebraic expression 3:06

Simplifying Variable Expressions Link: Combining Like Terms 2 Khan Academy video example of combining like terms to simplify a complex variable expression Reviews commutative property of addition to group like terms by changing the order 3:43

Link: Equivalent Forms of Expressions 1
Simplifying Variable Expressions Link: Equivalent Forms of Expressions 1 Khan Academy video with multiple choice examples using properties of operations to find simpler equivalent forms of algebraic expressions Reviews the distributive and commutative properties and explains why some of the expressions are not equivalent 5:32

Link: Equivalent Forms of Expressions 1
Simplifying Variable Expressions Link: Equivalent Forms of Expressions 1 Khan Academy interactive practice problems on simplifying algebraic expressions using distributive and commutative properties and combining like terms A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Polynomial Expressions
CONTENTS: Polynomial Expressions Introduction to polynomial expressions, including terminology, simplifying, adding and subtracting Definitions and Terminology Combining Like Terms to Simplify Polynomials Adding and Subtracting Polynomials Interactive Practice Problems: Adding and Subtracting Polynomials Operations with Polynomials Calculator Return to Topics List

Polynomial Expressions Link: Polynomials: Definitions / Evaluation Purplemath text lesson on the terminology of polynomials Includes definitions of term, degree, coefficient, constant term, leading term, polynomial, and monomial Also introduces evaluation of a polynomial at a specific number simplifying by combining like terms

Link: Polynomial Expressions 1 - Explaining Polynomial Expressions
Tutortag video introducing the terminology of polynomials Includes definitions of coefficient, polynomial, monomial, binomial, and trinomial Time 2:43

Link: Terms, Coefficients, and Exponents in a Polynomial
Polynomial Expressions Link: Terms, Coefficients, and Exponents in a Polynomial Khan Academy video introducing the terminology of polynomials via an example Includes identification of terms, coefficients, and exponents Time 2:36

Link: Ex: Intro to Polynomials in One Variable
Polynomial Expressions Link: Ex: Intro to Polynomials in One Variable Mathispower4u video introducing the terminology of polynomials via an example Includes definition of polynomial and identification of terms, degrees, coefficients, leading term, and the degree of a polynomial Time 2:59

Polynomial Expressions Link: Combining Like Terms Mathispower4u video introducing like terms and simplifying polynomials by combining like terms Includes definition of like terms and examples of like terms and unlike terms Discusses the distributive law and its role in combining like terms, as well as how combining like terms is done in practice Includes examples of simplifying polynomials in one or two variables Time 4:49

Polynomial Expressions Link: Simply a Polynomial Khan Academy video example of simplifying a polynomial by combining like terms Explains the difference between like and unlike terms Time 3:37

Polynomial Expressions Link: Addition and Subtraction of Polynomials Khan Academy video introducing addition and subtraction of polynomials by combining like terms Includes an explanation of what a polynomial is, plus terminology such as term, constant term, degree, coefficient, and standard form Examples include addition of polynomials in one variable by combining like terms subtraction of polynomials in two variables by first distributing the minus sign and then combining like terms constructing polynomials representing areas of figures Time 15:59

Polynomial Expressions Link: Simplifying Algebraic Expressions Mathispower4u video introducing using the distributive property to clear parentheses and combining like terms in polynomial expressions Reviews the distributive property Reviews multiplicative property of -1 (i.e., -1∙a = -a) and the opposite of a sum (i.e., distributing a minus sign across a sum) Includes examples of simplifying expressions Time 5:24

Mathispower4u video examples of simplifying polynomials Examples include adding and subtracting polynomials by clearing parentheses and combining like terms distributing a constant across a sum to clear parentheses distributing a non-constant monomial across a sum Time 4:22

Polynomial Expressions Link: Adding and Subtracting Polynomials Purplemath text lesson on adding and subtracting polynomials Includes examples of adding polynomials (both horizontally and vertically) by clearing parentheses and combining like terms subtracting polynomials (both horizontally and vertically) by distributing the minus sign (or -1) across a sum to clear parentheses and then combining like terms

Polynomial Expressions Link: How Do You Add Polynomials? Virtual Nerd video example of adding polynomials Gives step-by-step instructions to rewrite the sum without parentheses identify the like terms group like terms combine like terms rewrite the sum in standard form Time 4:59

Polynomial Expressions Link: Adding Polynomials Mathispower4u video examples of adding polynomials by clearing parentheses and combining like terms Time 5:09

Link: How Do You Subtract Polynomials?
Polynomial Expressions Link: How Do You Subtract Polynomials? Virtual Nerd video example of subtracting polynomials Gives step-by-step instructions to distribute the minus sign to remove parentheses identify the like terms group like terms combine like terms Time 2:07

Polynomial Expressions Link: Subtracting Polynomials Mathispower4u video examples of subtracting polynomials by distributing a -1 to clear parentheses and combining like terms Time 4:43

Polynomial Expressions Link: Adding and Subtracting Polynomials 1 Khan Academy video example of adding polynomials by clearing parentheses and combining like terms Time 2:01

Polynomial Expressions Link: Adding and Subtracting Polynomials 2 Khan Academy video example of subtracting polynomials by distributing a -1 to clear parentheses and combining like terms Time 1:43

Polynomial Expressions Link: Adding and Subtracting Polynomials 3 Khan Academy video example of adding and subtracting polynomials in one expression by clearing parentheses and combining like terms Time 2:27

Polynomial Expressions Link: Practice Adding/Subtracting Polynomials Khan Academy interactive practice problems on simplifying sums and differences of polynomials A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Polynomial Expressions Link: Operations with Polynomials Calculator Math Portal polynomial operation calculator Enter two polynomials and the calculator will add, subtract, multiply, or divide the polynomials as requested Calculator can also generate examples A complete step-by-step solution is given if requested

Polynomial Multiplication
CONTENTS: Polynomial Multiplication Description and examples of multiplying polynomials, including special products Multiplying Monomials Multiplying Monomials by Polynomials Multiplying Binomials; FOIL Method Special Products of Binomials: Squares and Conjugates Multiplying a Binomial and a Trinomial General Multiplication of Polynomials Interactive Practice Problems: Multiplying Simple Polynomials Interactive Practice Problems: Multiplying General Polynomials Operations with Polynomials Calculator Return to Topics List

Polynomial Multiplication Link: Multiplying Monomials Mathispower4u video examples of multiplying monomials and raising a monomial to a power using the laws of exponents Time 2:53

Link: How Do You Multiply Monomials?
Polynomial Multiplication Link: How Do You Multiply Monomials? Virtual Nerd video example simplifying a product of three monomials in two variables by regrouping the factors and using laws of exponents Time 1:20

Polynomial Multiplication Link: Multiplying Monomials by Polynomials Khan Academy video example of multiplying a monomial by a trinomial, using the distributive property and laws of exponents Time 2:43

Link: How Do You Multiply a Monomial by a Polynomial?
Polynomial Multiplication Link: How Do You Multiply a Monomial by a Polynomial? Virtual Nerd video example of multiplying a monomial by a trinomial in two variables, using the distributive property and laws of exponents Time 1:27

Link: Ex: Multiplying Using the Distributive Property
Polynomial Multiplication Link: Ex: Multiplying Using the Distributive Property Mathispower4u video examples of multiplying monomials by polynomials, using the distributive property and laws of exponents Time 2:54

Purplemath text lesson introducing multiplication of polynomials Topics and examples include multiplication of monomials using laws of exponents multiplication of a monomial by a polynomial using the distributive law and laws of exponents multiplication of two binomials horizontally using the distributive law, vertically, and using the “FOIL” method multiplication of polynomials with more than two terms horizontally and vertically

Polynomial Multiplication Link: FOIL for Multiplying Binomials Khan Academy video example of multiplying two binomials, first using the “FOIL” method and then using the distributive law twice Time 5:47

Polynomial Multiplication Link: Multiplying Binomials Mathispower4u video examples of multiplying two binomials using the distributive property and the “FOIL” method Time 4:12

Polynomial Multiplication Link: Special Products of Binomials Khan Academy video on squaring binomials and the product of a sum and a difference of two terms Topics include both specific examples and general formulas for the square of a binomial the product of the sum of two terms and the difference of the same two terms Time 10:36

Polynomial Multiplication Link: Ex: Squaring Binomials Mathispower4u video examples of squaring binomials by using the standard methods for multiplying binomials Also introduces formulas for squaring binomials without examples Time 3:22

Link: Ex: Multiplying Binomial Conjugates - Special Binomial Products
Polynomial Multiplication Link: Ex: Multiplying Binomial Conjugates - Special Binomial Products Mathispower4u video examples of the product of the sum of two terms and the difference of the same two terms using the standard methods for multiplying binomials Also defines conjugates and introduces the formula for a product of conjugates without examples Time 2:54

Link: Special Products of Polynomials 1
Polynomial Multiplication Link: Special Products of Polynomials 1 Khan Academy video on the product of the sum of two terms and the difference of the same two terms Derives and explains a general formula for such a product Includes an example using the formula Time 2:29

Link: Ex: Polynomial Multiplication Involving Binomials and Trinomials
Mathispower4u video examples of multiplying polynomials Examples are product of three factors: a monomial and two binomials product of a binomial and a trinomial Time 3:35

Polynomial Multiplication Link: Multiplying Polynomials Khan Academy video example multiplying a binomial by trinomial first horizontally using the distributive law twice, and then vertically Time 5:46

Polynomial Multiplication Link: More Multiplying Polynomials Khan Academy video example multiplying a binomial by trinomial using the distributive law twice Time 2:34

Polynomial Multiplication Link: Multiplying Polynomials 3 Khan Academy video example multiplying a binomial by trinomial using the distributive law twice The product of the polynomials is introduced as representing a volume Time 4:19

Polynomial Multiplication Link: Multiplication of Polynomials Khan Academy video examples of multiplying various types of polynomials Types of examples include monomial times polynomial binomial times binomial binomial times trinomial, multiplied vertically Time 9:49

Polynomial Multiplication Link: Multiplying Polynomials 1 Khan Academy interactive practice problems on multiplying simple polynomials A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Polynomial Multiplication Link: Multiplying Polynomials 2 Khan Academy interactive practice problems on multiplying more complicated polynomials A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Polynomial Multiplication Link: Operations with Polynomials Calculator Math Portal polynomial operation calculator Enter two polynomials and the calculator will add, subtract, multiply, or divide the polynomials as requested Calculator can also generate examples A complete step-by-step solution is given if requested

Polynomial Division CONTENTS:
Description and examples of dividing polynomials, including long division and synthetic division Polynomial Divided by a Monomial Long Division of Polynomials Synthetic Division Operations with Polynomials Calculator Return to Topics List

Link: Polynomial Divided by a Monomial
Polynomial Division Link: Polynomial Divided by a Monomial Khan Academy video example dividing a polynomial by a monomial using the distributive property and laws of exponents Time 4:44

Link: Ex: Divide a Polynomial by a Monomial
Polynomial Division Link: Ex: Divide a Polynomial by a Monomial Mathispower4u video examples dividing a polynomial by a monomial using the distributive property and laws of exponents Time 2:43

Purplemath text lesson on dividing polynomials Topics and examples include Dividing a polynomial by a monomial Factoring and “cancelling” (reducing a fraction) Long division, with and without a remainder

Mathispower4u video of long division of a polynomial by a binomial Includes review of long division of numbers written procedure for long division of polynomials explained with an example further examples dividing a polynomial by a binomial Time 9:25

Khan Academy video examples of long division of polynomials with remainders Time 12:09

Polynomial Division Link: Algebraic Long Division Khan Academy video with a (long) example of long division of polynomials Time 10:21

Link: How to do Long Division with Polynomials
Polynomial Division Link: How to do Long Division with Polynomials Mathbff video examples of long division of polynomials with remainders Includes polynomials with “missing” terms Time 11:13

Link: Divide a Polynomial by a Degree Two Binomial
Polynomial Division Link: Divide a Polynomial by a Degree Two Binomial Mathispower4u video example of long division of a polynomial by a degree two binomial with a remainder Time 4:14

Polynomial Division Link: Dividing Polynomials with Remainders Khan Academy video example of long division of polynomials with a remainder Includes a dividend with a “missing” term (i.e., 0 coefficient) Includes checking the answer by multiplying and simplifying Time 6:23

Mathispower4u video introducing synthetic division of a polynomial by a binomial of the form x – k Includes general explanation of the method of synthetic division several explicit examples example where the coefficient of x in the divisor is not 1 Time 10:07

Polynomial Division Link: Synthetic Division Khan Academy video explaining synthetic division of a polynomial by a binomial of the form x + k via an example Time 5:21

Link: How to Do Synthetic Division
Polynomial Division Link: How to Do Synthetic Division Mathbff video explaining synthetic division of a polynomial by a binomial of the form x - k via an example Time 6:35

Polynomial Division Link: Synthetic Division, Example 2 Khan Academy video example of synthetic division of a polynomial by a binomial of the form x - k Includes a dividend with a “missing” term (i.e., 0 coefficient) Time 5:00

Link: Ex 2: Divide a Polynomial by a Binomial Using Synthetic Division
Polynomial Division Link: Ex 2: Divide a Polynomial by a Binomial Using Synthetic Division Mathispower4u video example of synthetic division of a polynomial by a binomial of the form x + k Time 3:22

Link: Ex 3: Divide a Polynomial by a Binomial Using Synthetic Division
Polynomial Division Link: Ex 3: Divide a Polynomial by a Binomial Using Synthetic Division Mathispower4u video example of synthetic division of a polynomial by a binomial of the form x - k Includes a dividend with a “missing” term (i.e., 0 coefficient) Time 3:13

Link: Ex 4: Divide a Polynomial by a Binomial Using Synthetic Division
Polynomial Division Link: Ex 4: Divide a Polynomial by a Binomial Using Synthetic Division Mathispower4u video example of synthetic division of a polynomial by a binomial of the form x + k Includes a dividend with a “missing” term (i.e., 0 coefficient) Time 3:17

Polynomial Division Link: Operations with Polynomials Calculator Math Portal polynomial operation calculator Enter two polynomials and the calculator will add, subtract, multiply, or divide the polynomials as requested Calculator can also generate examples A complete step-by-step solution is given if requested

Factoring Polynomials
CONTENTS: Factoring Polynomials Description and examples of factoring polynomials, including special forms and factoring by grouping Greatest Common Factor of Monomials Factoring the GCF of Terms Out of a Polynomial Factoring a Difference of Squares Factoring a Sum or Difference of Cubes Factoring Perfect Square Trinomials Factoring Quadratic Polynomials and Trinomials Interactive Practice Problems: Factoring Quadratics with Leading Coefficient 1 Interactive Practice Problems: Factoring Quadratics with Leading Coefficient Not 1 Factoring by Grouping Interactive Practice Problems: Factoring by Grouping Return to Topics List

Link: How Do You Find the Greatest Common Factor of Monomials?
Factoring Polynomials Link: How Do You Find the Greatest Common Factor of Monomials? Virtual Nerd video example finding the greatest common factor of 4 monomials in 2 variables Steps used in the example to find the GCF are factor each monomial identify common factors list the common factors multiply the common factors Time 3:52

Factoring Polynomials Link: Monomial Greatest Common Factor Khan Academy video example finding the greatest common factor of 2 monomials in 2 variables Time 3:13

Link: Factor Expressions Using the GCF
Factoring Polynomials Link: Factor Expressions Using the GCF Khan Academy video example of factoring the greatest common factor of the terms out of a binomial in two variables Begins with a review of finding the GCF of two monomials Time 5:28

Link: Ex 1: Identify GCF and Factor a Binomial
Factoring Polynomials Link: Ex 1: Identify GCF and Factor a Binomial Mathispower4u video examples of factoring the greatest common factor of the terms out of binomials Includes finding the GCF of the terms Time 4:16

Link: Ex 2: Identify GCF and Factor a Trinomial
Factoring Polynomials Link: Ex 2: Identify GCF and Factor a Trinomial Mathispower4u video examples of factoring the greatest common factor of the terms out of trinomials Includes finding the GCF of the terms Time 6:40

Factoring Polynomials
Link: How Do You Factor the Greatest Common Factor Out of a Polynomial? Virtual Nerd video example of factoring the greatest common factor of the terms out of a polynomial in two variables Includes finding the GCF of the terms Time 4:19

Factoring Polynomials Link: Simple Polynomial Factoring Purplemath text lesson on basic polynomial factoring techniques Topics and examples include factoring out the greatest common factor of the terms factoring by grouping (referred to here as “factoring in pairs”) for quadratics

Link: How Do You Know if You Have a Difference of Squares?
Factoring Polynomials Link: How Do You Know if You Have a Difference of Squares? Virtual Nerd video explaining how to identify a binomial as a “difference of squares’’ Two issues to be aware of: considers only binomials with perfect square coefficients the condition “the degree of each term must be even or 0” is true but slightly misleading, since 0 is even Time 2:26

Link: How Do You Factor a Polynomial Using Difference of Squares?
Factoring Polynomials Link: How Do You Factor a Polynomial Using Difference of Squares? Virtual Nerd video example giving a step-by-step procedure to factor a binomial that is a difference of squares Time 4:45

Link: Factoring a Difference of Squares
Factoring Polynomials Link: Factoring a Difference of Squares Mathispower4u video introducing the factoring of a difference of squares Video includes identifying a difference of squares general formula for the factors of a difference of squares numerous examples from very basic to somewhat more advanced Time 5:25

Link: Example: Factor a Difference of Squares
Factoring Polynomials Link: Example: Factor a Difference of Squares Mathispower4u video examples of factoring difference of squares binomials Includes review of the formula for the factors of a difference of squares and several examples Time 4:19

Link: Ex: Factor a Difference of Squares with a Common Factor
Factoring Polynomials Link: Ex: Factor a Difference of Squares with a Common Factor Mathispower4u video example of factoring a difference of squares binomial in which the terms have a (perfect square) common factor Time 3:06

Link: Factoring a Sum or Difference of Cubes
Factoring Polynomials Link: Factoring a Sum or Difference of Cubes Mathispower4u video introducing factoring of sum of cubes and difference of cubes binomials Video includes general formulas for factors of sum and difference of cubes binomials verification of sum of cubes formula using long division examples, including one in which the terms of the binomial have a common factor Time 6:07

Link: How Do You Factor a Polynomial Using a Sum of Cubes?
Factoring Polynomials Link: How Do You Factor a Polynomial Using a Sum of Cubes? Virtual Nerd video example giving a step-by-step procedure for factoring a sum of cubes binomial Time 4:15

Link: How Do You Factor a Polynomial Using a Difference of Cubes?
Factoring Polynomials Link: How Do You Factor a Polynomial Using a Difference of Cubes? Virtual Nerd video example giving a step-by-step procedure for factoring a difference of cubes binomial Time 4:40

Link: Ex 1: Factor a Sum or Difference of Cubes
Factoring Polynomials Link: Ex 1: Factor a Sum or Difference of Cubes Mathispower4u video examples of factoring a sum of cubes binomial and a difference of cubes binomial Video includes brief review of formulas for factors of sum and difference of cubes a basic example of each type Time 3:19

Link: Ex 2: Factor a Sum or Difference of Cubes
Factoring Polynomials Link: Ex 2: Factor a Sum or Difference of Cubes Mathispower4u video examples of factoring a sum of cubes binomial and a difference of cubes binomial Video includes brief review of formulas for factors of sum and difference of cubes an example of each type in which the terms of the binomial have a common factor Time 3:35

Link: Ex 3: Factor a Sum or Difference of Cubes
Factoring Polynomials Link: Ex 3: Factor a Sum or Difference of Cubes Mathispower4u video example of factoring a sum of cubes binomial Video includes brief review of formulas for factors of sum and difference of cubes an example of a sum of cubes in which the terms of the binomial have a perfect cube common factor and relatively large coefficients Time 5:23

Factoring Polynomials
Link: Factoring Perfect Square Trinomials & the Difference of Two Squares AlRichards314 video on factoring trinomials that are squares of binomials and factoring difference of squares binomials Video includes review of formulas for the square of a binomial and the product of a sum of two terms and the difference of the same terms product formulas reinterpreted as factoring formulas several examples of each type Time 7:45

Link: Factoring a Perfect Square Trinomial
Factoring Polynomials Link: Factoring a Perfect Square Trinomial Mathispower4u video on factoring trinomials that are squares of binomials Video includes how to recognize a perfect square trinomial formula for the factors of a perfect square trinomial several examples, including one in which the terms of the trinomial have a common factor Time 7:52

Link: How Do You Use a Shortcut to Factor a Perfect Square Trinomial?
Factoring Polynomials Link: How Do You Use a Shortcut to Factor a Perfect Square Trinomial? Virtual Nerd video on factoring trinomials that are perfect squares of binomials Video includes how to recognize the pattern of a perfect square trinomial formula for the factors of a perfect square trinomial (the “shortcut”) an example, using the formula and a step-by-step procedure Time 8:10

Link: Ex: Factor a Perfect Square Trinomial
Factoring Polynomials Link: Ex: Factor a Perfect Square Trinomial Mathispower4u video examples of the special case of factoring quadratic polynomials that are perfect square trinomials Does not rely on memorized formula for factoring; instead uses a modified “trial and error” method given that the degree 2 and constant terms are perfect squares Time 5:33

Factoring Polynomials Link: Factoring Quadratics Purplemath text lesson on factoring quadratic polynomials Topics and examples include the “simple case” where the leading coefficient is 1 the “hard case” where the leading coefficient is not 1, using both the “trial and error” method and the “box method” the “weird case” of quadratic type polynomials that are not actually quadratic

Factoring Polynomials Link: Factoring Trinomials with a Leading 1 Khan Academy video example of factoring a quadratic polynomial with leading coefficient 1 Video includes an explanation of what to look for (and why) in the coefficients of the quadratic in order to find the factors Method is illustrated with an example Time 4:12

Link: Factoring a Trinomial with Leading Coefficient of 1 - The Basics
Factoring Polynomials Link: Factoring a Trinomial with Leading Coefficient of 1 - The Basics Mathispower4u video examples of factoring a quadratic polynomial with leading coefficient 1 Video includes an explanation of what to look for (and why) in the coefficients of the quadratic in order to find the factors Method is illustrated with numerous examples, including one in which the terms of a trinomial have a common factor Time 10:34

Factoring Polynomials Link: Factoring Quadratic Expressions Khan Academy video examples of factoring a quadratic polynomial with leading coefficient 1 Video includes an explanation of what to look for (and why) in the coefficients of the quadratic in order to find the factors Method is illustrated with many examples, including one in which the terms of a trinomial have a common factor Time 16:30

Link: Factoring Trinomials: Trial and Error and Grouping
Factoring Polynomials Link: Factoring Trinomials: Trial and Error and Grouping Mathispower4u video examples of factoring a quadratic polynomial where the leading coefficient is not 1 Video includes trial and error (or reverse FOIL) method with an explanation of the procedure and several examples grouping method with several examples Time 9:45

Factoring Polynomials Link: Factoring ax^2+bx+c (Part 1) AlRichards314 video examples of factoring a quadratic polynomial where the leading coefficient is not 1 Method used is a combination of trial and error (or reverse FOIL) and grouping Part 2 includes more examples Time 5:22

Factoring Polynomials Link: Factoring ax^2+bx+c (Part 2) AlRichards314 video examples of factoring a quadratic polynomial where the leading coefficient is not 1 Method used is a combination of trial and error (or reverse FOIL) and grouping Continuation of Part 1 with more examples Time 4:49

Link: Factoring Trinomials with a Common Factor
Factoring Polynomials Link: Factoring Trinomials with a Common Factor Khan Academy video example of factoring a quadratic polynomial in which the terms have a numerical common factor Method used is a combination of trial and error (or reverse FOIL) and grouping Time 5:01

Factoring Polynomials Link: Practice: Factoring Quadratics 1 Khan Academy interactive practice problems on factoring quadratic polynomials with leading coefficient 1 A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Factoring Polynomials Link: Practice: Factoring Quadratics 2 Khan Academy interactive practice problems on factoring quadratic polynomials where the leading coefficient is not 1 A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Factoring Polynomials Link: Factor by Grouping Mathispower4u video examples of factoring polynomials using the grouping method Video includes introduction to the grouping method via an example written description of the steps in the grouping method several examples of factoring cubic polynomials by grouping example using grouping to solve a quadratic equation by factoring Time 7:48

Link: Ex: Factor a Polynomial Using the Grouping Technique
Factoring Polynomials Link: Ex: Factor a Polynomial Using the Grouping Technique Mathispower4u video examples of factoring cubic polynomials using the grouping method Time 3:51

Link: Factoring by Grouping and Factoring Completely
Factoring Polynomials Link: Factoring by Grouping and Factoring Completely Khan Academy video examples of factoring quadratic polynomials using the grouping method Includes an example of a trinomial in which the terms have a common factor Time 13:57

Factoring Polynomials Link: Example: Basic Grouping Khan Academy video example of factoring an expression with 4 terms in 2 variables using the grouping technique Time 3:46

The Cool Math Guy video examples of factoring polynomials Video includes factoring out the greatest common factor of the terms factoring quadratic polynomials factoring by grouping Time 9:56

Link: Practice: Factoring Polynomials by Grouping
Khan Academy interactive practice problems on factoring polynomials using the grouping technique A sequence of five practice problems is provided, along with help with correct formats to enter answers hints to help solve the problem, if needed links to video lessons if you get stuck

Formulas & Polynomials in Two (or more) Variables
CONTENTS: Formulas & Polynomials in Two (or more) Variables Description and examples of formulas and polynomial equations, including evaluation at specific numbers and solving for one variable in terms of the others Equations and Formulas Writing a Polynomial Expression from a Verbal Phrase Evaluating Formulas and Polynomials for Specific Values of the Variables Solving for One Variable in Terms of the Others in a Formula Interactive Practice Problems: Solving Equations in Terms of a Variable Return to Topics List

Formulas & Polynomials in Two (or more) Variables Link: Equations and Formulas Math Is Fun text lesson on equations, formulas, and the difference between them

Formulas & Polynomials in Two (or more) Variables Link: Example Question Involving Polynomials Tutortag video example of translating a verbal phrase into a polynomial expression Example expresses the difference in areas between two geometric figures whose dimensions are given as variables Time 5:34

Link: Evaluate a Formula Using Substitution
Formulas & Polynomials in Two (or more) Variables Link: Evaluate a Formula Using Substitution Khan Academy video example using substitution in a formula to convert degrees Celsius to degrees Fahrenheit Time 1:15

Link: Evaluating a Polynomial at a Given Value
Formulas & Polynomials in Two (or more) Variables Link: Evaluating a Polynomial at a Given Value Khan Academy video example of substitution of a negative number in a quadratic polynomial expression Time 1:09

Formulas & Polynomials in Two (or more) Variables
Link: How Do You Evaluate an Algebraic Expression by Plugging in Values? Virtual Nerd video example of evaluating a polynomial expression in two variables at given values of the variables Time 1:07

Link: Polynomial Expressions and Polynomial Functions
Formulas & Polynomials in Two (or more) Variables Link: Polynomial Expressions and Polynomial Functions Mathispower4u video examples of substitution of numbers in polynomial expressions Video includes review of polynomial terminology examples of evaluation of polynomials in more than one variable at specific values of the variables evaluation of polynomial expressions in one variable at specific numbers both by hand and using a graphing calculator Time 7:25

Formulas & Polynomials in Two (or more) Variables Link: Evaluate a Quadratic Equation - Death Application Mathispower4u video example evaluating a quadratic polynomial at a number Polynomial formula models a real-world problem and includes non-integer (decimal) coefficients A calculator is used for most of the calculations Time 4:52

Link: Evaluate a Polynomial Equation - Flu Virus Application
Formulas & Polynomials in Two (or more) Variables Link: Evaluate a Polynomial Equation - Flu Virus Application Mathispower4u video example evaluating a fourth degree polynomial at a number Polynomial formula models a real-world problem and includes non-integer (decimal) coefficients A calculator is used for some of the calculations Time 4:01

Formulas & Polynomials in Two (or more) Variables Link: Solving Equations and Formulas Slide Share text slides with examples of formulas in more than one variable and solving for one variable in terms of the others Emphasizes “undoing” operations to isolate the desired variable

Formulas & Polynomials in Two (or more) Variables Link: Solving Literal Equations Purplemath text lesson on solving for one variable in terms of the others in a formula with two or more variables Topics and examples include geometric and other types of formulas in three or more variables rewriting a linear equation in slope-intercept form to find the slope of the line

Link: Rearrange Formulas to Isolate Specific Variables
Formulas & Polynomials in Two (or more) Variables Link: Rearrange Formulas to Isolate Specific Variables Khan Academy video example solving the area of a triangle formula for the height in terms of the area and base Time 2:29

Formulas & Polynomials in Two (or more) Variables Link: Solving for a Variable Khan Academy video example solving the rectangle perimeter formula for the length in terms of the perimeter and width Time 1:24

Link: Example: Solving for a Variable
Formulas & Polynomials in Two (or more) Variables Link: Example: Solving for a Variable Khan Academy video example solving the rectangle perimeter formula for the width in terms of the perimeter and length Problem is solved in various ways Time 6:58

Link: Solving for a Variable 2
Formulas & Polynomials in Two (or more) Variables Link: Solving for a Variable 2 Khan Academy video example solving for the Fahrenheit temperature in the formula giving the Celsius temperature in terms of the Fahrenheit temperature Time 2:41

Link: Solving Equations in Terms of a Variable
Formulas & Polynomials in Two (or more) Variables Link: Solving Equations in Terms of a Variable Khan Academy interactive practice problems on solving for one variable in terms of the others in an equation in two or more variables A sequence of five practice problems is provided, along with hints to help solve the problem, if needed links to video lessons if you get stuck

Polynomial Functions CONTENTS:
Description and examples of the behavior of polynomial functions, including end behavior, smoothness, turning points, and graphs End Behavior of Polynomial Function General Properties of Polynomial Functions Return to Topics List

Link: Summary of End Behavior of Polynomial Function
Polynomial Functions Link: Summary of End Behavior of Polynomial Function Mathispower4u video describing the end (or long run) behavior of a polynomial in terms of the degree and the sign of the leading coefficient Time 5:48

Polynomial Functions Link: How Polynomials Behave Math Is Fun text lesson on the behavior of the graphs of polynomial functions Topics and examples include continuity and smoothness behavior of xn for even and odd values of n behavior of axn for various values of a turning points end behavior

Polynomial Functions Link: Polynomial Graphs: General Behavior Purplemath text lesson on the behavior of the graphs of polynomial functions Topics and examples include end behavior zeroes and their multiplicities degrees and turning points quick graphing