C ollege A lgebra Basic Algebraic Operations (Appendix A)

2 Objectives Appendix A Cover the topics in Appendix A (A1 - A 4): A-1 : Algebra and real numbers. A-2 : Polynomials : basic operations.

3 A-1:Algebra and real numbers Appendix A A-1 : contents. 1 : The Set of Real Numbers 2 : Subsets of the Set of Real Numbers 3 : The Real Number Line 4 : Basic Real Number Properties. 5:Further Real Number Properties

NNatural Numbers 1, 2, 3,... ZIntegers..., –2, –1, 0, 1, 2,... Q Rational Numbers –4, 0, 8, –3 5, 2 3,– _ IIrrational Numbers 2,  3 7, RReal Numbers –7, 0, 3 5, –2 3, 3.14, –,  1.The Set of Real Numbers A-1 SymbolName _ DescriptionExamples Positive integers Natural numbers, their negatives and 0 Numbers, can be represented as a/b a, and b integers and b≠0 Decimal are repeating or terminating Numbers, can be represented as non repeating nor terminating Rational and irrational numbers

2.Subsets of the Set of Real Numbers Natural numbers (N) Negatives of natural numbers Zero Integers (Z) Noninteger ratios of integers Rational numbers (Q) Irrational numbers (I) Real numbers (R) N  Z  Q  R A-1

6 3.The Real Number Line A-1 Origin Positive real numbersNegative real numbers

4.Basic Real Number Properties Let R be the set of real numbers and let x, y, and z be arbitrary elements of R. Addition Properties Closure:x +y is a unique element in R. Associative:(x +y ) + z = x + ( y + z ) Commutative:x +y =y +x Identity:0 +x =x + 0 = x Inverse:x + (– x ) = (– x ) + x = 0 A-1

Multiplication Properties Closure: xy is a unique element in R. Associative: Commutative: xy =yx Identity: (1) x = x (1) = x Inverse: X       1 x =       1 x x = 1x  0 Combined Property Distributive: x ( y + z) = xy + xz ( x + y ) z = + yz (xy)z = x(yz) A-1 5.Basic Real Number Properties-cont.

9 A-1 6.Further Real Number Properties Subtraction and Division For all real numbers a and b Subtraction: a – b = a + (- b) Division : b/a = a ÷ b = a/ b = a(1/b) Division by ZERO not allowed Properties for negative integers: 1.-(-a) = a 2.(-a) b = -(a b) = a(-b)= -a b 3.(-a)(-b)=a b 4.(-1)a = -a 5.-a/-b = -(a/-b) = -(-a/b) = a/b b≠0 6.-a/b = -(a/b) = (a/-b) b≠0 Zero Properties : 1.a * 0 = 0 2.a* b = 0 if and only if a=0 or b=0 or both

10 A-2 : Polynomials : Basic Operations Appendix A A-2 : After completing this tutorial, you should be able to: 1.Identify a term, coefficient, constant term, and polynomial. 2.Tell the difference between a monomial, binomial, and trinomial. 3.Find the degree of a term and polynomial. 4.Combine like terms. 5.Add and subtract polynomials. 6.Multiply any polynomial times any other polynomial. 7.Use the FOIL method to multiply a binomial times a binomial. 8.Use special product rules to multiply a binomial squared and a product of a sum and difference of two terms.

11 A-2 A-2 : Polynomials Algebraic Expressions Formed by using constants and variables and the algebraic operations of addition,subtraction, multiplication, division, rising to power, and taking roots Examples: Polynomial An algebraic Expressions involving only the operations of addition,subtraction, multiplication, rising to natural number power on constants and variables Examples: (Degree of the term is the sum of powers of variables) (Degree of the whole polynomial is the highest degree in the polynomial)

12 A-2 A-2 : Polynomials. Polynomials and non Polynomials A)Polynomial in one variable ( Degree of the whole polynomial = 6) B) Polynomial in several variable ( Degree of the whole polynomial = 3) C) Non-Polynomial

13 A-2 A-2 : Polynomials. Term A term is a number, variable or the product of a number and variable(s). Examples of terms are, z. Coefficient A coefficient is the numeric factor of your term. Here are the coefficients of the terms listed above: TermCoefficient z1

14 A-2 A-2 : Polynomials. Constant Term A constant term is a term that contains only a number. In other words, there is no variable in a constant term. Examples of constant terms are 4, 100, and -5. Standard Form of a Polynomial where n is a non-negative integer. is called the leading coefficient. is a constant. In other words, a polynomial is a finite sum of terms where the exponents on the variables are non-negative integers. Note that the terms are separated by +’s and -‘s.

15 A-2 A-2 : Polynomials. Degree of a Term The degree of a term is the sum of the exponents on the variables contained in the term. For example, the degree of the term would be = 2. The exponent on a is 1 and on b is 1 and the sum of the exponents is 2. The degree of the term would be 3 since the only variable exponent that we have is 3.

16 A-2 A-2 : Polynomials. Degree of the Polynomial The degree of the polynomial is the largest degree of all its terms Descending Order Note that the standard form of a polynomial that is shown above is written in descending order. This means that the term that has the highest degree is written first, the term with the next highest degree is written next, and so forth.

17 A-2 A-2 : Polynomials. Some Types of Polynomials TypeDefinitionExample Monomial A polynomial with one term 5x5x Binomial A polynomial with two terms 5x - 10 Trinomial A polynomial with three terms

18 A-2 A-2 : Polynomials. Example 1: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: Since the degree of the polynomial is the highest degree of all the terms, it looks like the degree is 3. Since there are three terms, this is a trinomial.

19 A-2 A-2 : Polynomials. Example 2: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: Since the degree of the polynomial is the highest degree of all the terms, it looks like the degree is 3. Make sure that you don’t fall into the trap of thinking it is always the degree of the first term. This polynomial is not written in standard form (descending order). So we had to actually go to the second term to get the highest degree. Since there are two terms, this is a binomial.

20 A-2 A-2 : Polynomials. Example 3: Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these -20.: Since the degree of the polynomial is the highest degree of all the terms, it looks like the degree is 0. Since there is one term, this is a monomial.

21 A-2 A-2 : Polynomials. Combining Like Terms Recall that like terms are terms that have the exact same variables raised to the exact same exponents. One example of like terms is. Another example is. You can only combine terms that are like terms. You can think of it as the reverse of the distributive property. It is like counting apples and oranges. You just count up how many variables you have the same and write the number in front of the common variable part.

22 A-2 A-2 : Polynomials. Adding and Subtracting Polynomials Step 1: Remove the ( ). If there is only a + sign in front of ( ), then the terms inside of ( ) remain the same when you remove the ( ). If there is a - in front of the ( ), then distribute it by multiplying every term in the ( ) by a -1 (or you can think of it as negating every term in the ( )). Step 2: Combine like terms.

23 A-2 A-2 : Polynomials. Example 4: Perform the indicated operation and simplify: *Remove the ( ) *Add like terms together

24 A-2 A-2 : Polynomials. Example 5: Perform the indicated operation and simplify: *Dist. the - through second ( ) *Combine like terms

25 A-2 A-2 : Polynomials. Multiplying Polynomials In general, when multiplying two polynomials together, use the distributive property until every term of one polynomial is multiplied times every term of the other polynomial. Make sure that you simplify your answer by combining any like terms. On this page we will look at some of the more common types of polynomials to illustrate this idea. Multiplying Polynomials In general, when multiplying two polynomials together, use the distributive property until every term of one polynomial is multiplied times every term of the other polynomial. Make sure that you simplify your answer by combining any like terms. On this page we will look at some of the more common types of polynomials to illustrate this idea. (Monomial)(Monomial) In this case, there is only one term in each polynomial. You simply multiply the two terms together.

26 A-2 A-2 : Polynomials. Example 6: Find the product *Mult. like bases add exp.

27 A-2 A-2 : Polynomials. (Monomial)(Polynomial) In this case, there is only one term in one polynomial and more than one term in the other. You need to distribute the monomial to EVERY term of the other polynomial. Example 6: Find the product *Dist. -5b *Mult. like bases add exp.

28 A-2 A-2 : Polynomials. (Binomial)(Binomial) In this case, both polynomials have two terms. You need to distribute both terms of one polynomial times both terms of the other polynomial. One way to keep track of your distributive property is to Use the FOIL method. Note that this method only works on (Binomial)(Binomial). FFirst terms OOutside terms IInside terms LLast terms This is a fancy way of saying to take every term of the first binomial times every term of the second binomial. In other words, do the distributive property for every term in the first binomial.

29 A-2 A-2 : Polynomials. Example 7: Find the product *Use the FOIL method *Combine like terms

Foil Method FOIL FirstOuterInnerLast Product  (2x – 1)(3x + 2) = 6x 2 +4x – 3x – 2 1.(a – b)( a +b ) = a 2 – b 2 2.(a +b) 2 =a 2 + 2ab +b 2 3.(a – b) 2 =a 2 – 2ab +b 2 A-2 Special Products 4 : Multiplication.

31 A-2 A-2 : Polynomials. Example 8: Find the product * Example 9: Find the product *

32 A-2 A-2 : Polynomials. Example 10: Find the product ** *Use Dist. Prop. twice *Combine like terms

33 A-2 A-2 : Polynomials. Special product rule for binomial cubed: ** In other words, when you have a binomial cubed, you end up with the first term cubed plus (or minus) three times the first term squared times the second term plus three times the first term times the second term squared plus (or minus) the last term cubed. Any time you have a binomial cubed you can use this shortcut method to find your product.

34 A-2 A-2 : Polynomials. Example 11: Find the product **