Chapter 5 – 5-1 Monomials Mon., Oct. 19 th Essential Question: Can you apply basic arithmetic operations to polynomials, radical expressions and complex numbers Target: Students will multiply and divide monomials Agenda: Test Chapter 5 – 5-1 Monomials In-class Chapter 5 Homework: Pg #19-41 odd

Chapter 5 – 5-1 Monomials Wed., Oct. 21 st Essential Question: Can you apply basic arithmetic operations to polynomials, radical expressions and complex numbers Target: Students will identify the differences between a monomial and polynomial and be able to simplify polynomials Agenda: Chapter 5-1 Homework: Pg #19-41 odd 5-2 Introduction 5-2 Homework (Pg. 231 #17-33 odd, odd)

Monomials An expression that is a number, a variable, or a product of a number and one or more variables Cannot contain variables in denominator, variables with exponents that are negative, or variables under radicals MonomialsNot Monomials 5b, -w, 23, x 2, 1/2x 3 y 4 1/n 2, √x, x+8, a -1

Terms Constants: Monomials that contain no variable (i.e., 23, -1) Coefficient: Numeric factor of a monomial (4x – coefficient is 4)

Terms Degree of a monomial: Sum of the exponents of its variables 4x 2 y 3 - degree is 5 Power: An expression of the form x n

Simplifying Expressions When exponents are multiplied – add their powers for the like variables: (3x 2 y3)(2xy 2 ) = = 3 ∙ x ∙ x ∙ y ∙ y ∙ y ∙ 2 ∙ x ∙ y ∙ y (expanded form) = 3 ∙ 2 ∙ x 3 ∙ y 5 (simplified) For any real number a and integer m and n, a m ∙ a n = a m + n

Simplifying Expressions When exponents are divided – subtract their powers for the like variables: (3x 2 y3)/(2xy 2 ) = 3 ∙ x ∙ x ∙ y ∙ y ∙ y (expanded form) 2 ∙ x ∙ y ∙ y = 3xy / 2 (simplified) For any real number a and integer m and n, a m / a n = a m-n

Simplifying Expressions When exponents are to a power – multiply their powers for the like variables: (3x 2 y3) 2 = = 3 2 ∙ x 2 ∙2 ∙ y 3 ∙ 2 = 9x 4 y 6 For any real number a and integer m and n, (a m ) n = a m ∙ n

Negative Exponent For any real number a ≠ 0 and any integer n, a -n = 1/a n and 1/a -n = a n therefore: a -4 = 1/a 4 And: 1/a -3 = a 3

5.2 POLYNOMIALS

Polynomials Definition: Polynomial is a monomial or a sum or monomials Monomials that make up polynomial are called TERMS of the polynomial X 2, xy, x, each are different TERMS (Variable AND exponents do not match – these CANNOT be combined) X 2, 4x 2, 21x 2 are all the same TERM (Variable AND exponent DO match – these CAN be combined)

Polynomials Variables: Terms of the unknown shown by a letter (x, y, ab, …) Coefficients: Numbers multiplied to variables Exponents: Repeated multiplication Leading Coefficient: The coefficient of the highest power Constant: The number with no variable

Polynomials Degree of Polynomial: Is the degree of the monomial with the greatest degree 5x 3 y 5 – 9x 4 - Degree is 8 Standard Form : Write polynomial terms from highest to lowest exponents (x 3 + x 2 + 5)

Simplifying Polynomials To simplify polynomials means to perform the operations and combine LIKE terms Add & Subtract – COMBINE like terms Multiply multiply coefficients add exponents to like variables Combine like terms Powers Raise coefficients to power Multiply exponents to like variables Combine like terms

Chapter 5 – Factoring Bootcamp Fri., Oct. 23 rd Essential Question: Can you apply basic arithmetic operations to polynomials, radical expressions and complex numbers Target: Students will simplify polynomial quotients by factoring Agenda: Pick up 5.1 homework 5-2 Homework (Pg. 231 #17-33 odd, odd) Quiz 5.1 & 5.2 Factoring Bootcamp Factoring Homework - Worksheet

Factoring Finding Greatest Common Factor (GCF) Difference of 2’s Factoring Trinomials with leading coefficients of 1 Factoring Trinomials with leading coefficients other than 1 Factoring 4 terms by grouping

5.4 FACTORING POLYNOMIALS Grouping, Trinomials, Binomials, GCF & Solving Equations

Factoring a polynomial means expressing it as a product of other polynomials. Factoring is a method to solve. To solve a polynomial, set it equal to zero. Factor it. Set each factor with a variable equal to zero and solve. This lesson focuses on methods to factor.

FACTORING POLYNOMIALS 1. Greatest Common Monomial Factor (GCF) STEPS TO FACTOR A POLYNOMIAL 2. Is it a Binomial? a. Difference of Two Squares (DOS) b. Difference of Two Cubes (DOC) c. Sum of Two Cubes (SOC) 3. Is it a Trinomial? a. Perfect Square Trinomial (PST) b. Guess Method 4. Four or more terms? a. Grouping

Factoring polynomials with a common monomial factor (using GCF). **Always look for a GCF before using any other factoring method. Factoring Method #1

Step 1: Step 2: Divide by GCF The answer should look like this:

Factor these on your own looking for a GCF.

A “Difference of Squares” is a binomial ( *2 terms only*) and it factors like this: Some examples of variable squares are Factoring Method #2

To factor a DOS, express each term as a square of a monomial then apply the rule...

Here is another example:

Try these on your own: Watch for GCFs and subsequent DOS.

Sum and Difference of Cubes: Another binomial factoring technique Note: the resulting quadratic is always prime and will provide two imaginary solutions! Do not factor it! Some examples of cubes are 1, 8, 27, 64, 125 and

Rewrite as cubes Write each monomial as a cube and apply either of the rules. Apply the rule for sum of cubes:

Rewrite as cubes Apply the rule for difference of cubes:

Factoring Method #3 Factoring a trinomial in the form that mimics: But looks like this instead

Factors of +8: 1 & 8 2 & 4 -1 & & -4 Since we still have squares we must See if we can still factor. This is a Difference of squares!

Lets do another example: Find a GCF Factor trinomial Always check for GCF before you do anything else. Since neither is a DOS, You are Done!

You try FOIL You can check with FOIL

Factoring Technique #4 Factoring By Grouping for polynomials with 4 or more terms 1. Group the first set of terms and last set of terms with parentheses. 2. Factor out the GCF from each group so that both sets of parentheses contain the same factors. 3. Factor out the GCF again (the GCF is the factor from step 2).

Step 1: Group Example 1: Step 2: Factor out GCF from each group Step 3: Factor out GCF again

Example 2: GCF Split in half GCF each half DOS

Try using factoring now to solve for x

Answers:

Factoring Completely Now that we’ve learned all the types of factoring, we need to remember to use them all. Whenever it says to factor, you must break down the expression into the smallest possible factors. Let’s review all the ways to factor.

Types of Factoring 1.Look for GCF first. 2.Count the number of terms: a) 4 terms – factor by grouping b) 3 terms - look for perfect square trinomial if not, factor using the BX term c) 2 terms - look for difference of squares, sum or difference of cubes If any ( ) still has an exponent of 2 or more, see if you can factor again.

Chapter 5 – Dividing Polynomials Thurs., Oct. 29 th Essential Question: Can you apply basic arithmetic operations to polynomials, radical expressions and complex numbers Target: Students will divide polynomials using long division Agenda: Pick up homework 5-4 Factoring Homework (pg. 236 – 5, 12, all, 33, add) Finish on Factoring Worksheet Quiz Factoring (CAN USE HOMEWORK) 5-3 – Dividing using Long Division BRING HEADPHONES on MONDAY

Chapter 5 – Review – 5.1 – 5.4 Nov. 4 th Essential Question: Can you apply basic arithmetic operations to polynomials, radical expressions and complex numbers Target: Students will review concepts learned in Chapter 5 Agenda: 5-3 – Day 2 Synthetic division Homework Review 5.1 – 5.2 Quiz Review Factoring Quiz Review Move into Group for Review Work Get Review Packet checked off for NOTE SHEET approval