CHAPTER 5: Polynomials and Polynomial Functions EQ: What are the different ways to factor polynomials?

Activation What is the difference between Term Coefficient Polynomial Take a minute and talk to your seat partner.

5-2 Addition and Subtraction of Polynomials EQ: 5-1 What is a polynomial? 5-2 How do you add and subtract polynomials? 5-1 Polynomials and Functions

Vocabulary Terms—a combination of numbers, variables and exponents combined through multiplication Examples: 3 x 2 4xy -6xz Coefficient—the numeric part of a term Degree of term—the sum of the exponents in a term Polynomial—two or more terms joined through addition or subtraction Degree of a polynomial—the highest degree of its terms

Vocabulary Monomial—one term Binomial—two terms Trinomial—three terms Ascending order—the degree of the terms gets larger going from left to right Descending order—the degree of the terms gets smaller going from left to right Like terms—terms with exactly the same variable lables

Vocabulary Standard form—placing a polynomial in order by degree from highest to lowest – X –linear – X 2 –quadratic – X 3 –cubic

Simplify the following polynomial: 3x – 4y + 7x – 2 + 5y 3x 2 y + 2xy 2 – 4x 2 y + 7x 2

Evaluate the following polynomial functions: Given:f(x) = 3x 2 + 2x – 7 Find:f(4) f(-1) 3 f(2)

Add the following polynomials vertical method (3x 2 – 2x + 5) + (4x 2 + 5x – 7)

Add the following polynomials horizontal method (3x 2 – 2x + 5) + (4x 2 + 5x – 7)

Subtract the following polynomials (3x 2 – 2x + 5) - (4x 2 + 5x – 7) Remember subtraction can be done using the additive inverse.

SUMMARIZATION

HOMEWORK PAGE(S): NUMBERS: 2 – 22 evens, 28, 32, 36 PAGE(S): NUMBERS: 2 – 20 evens, 24, 28, 34

Activation Multiply x 2 x 3

5-3 Multiplication of Polynomials EQ: How do you multiply two polynomials?

Multiply: (3x – 2)(2x + 5) 6x x – 4x – 10 6x 2 +11x - 10 FOILFOIL

(3x 2 – 2x + 5)(x – 4) How should we approach the problem with additional terms in one or more of the polynomials

(3x 2 – 2x + 5)(x – 4) Alternate Vertical Method

SUMMARIZATION 1. 2.

HOMEWORK PAGE(S): NUMBERS: 4, 8, 12, 16, 20, 24, 28, 34, 44

Activation Create a factor tree for: 3048 What we have done is the prime factorization of each number. So what are factors? Think back to elementary school

5-4 Factoring 5-5 More Factoring EQ: What is factoring?

0—Make sure the terms are in order x 2, x, constant Examples: x 2 – 3x 4 + 2x =______________________ 1—Look for a common factor May be a number or a variable or both Basic rules of factoring Examples: 4x 2 – 8 ________ 9x 3 y 2 +15x 2 y 2 – 3xy 2 _________________

2—if there are two terms remaining: Check to see if each is a perfect square or a perfect cube and follow the formulas Difference of two squares (must be joined by - ) a 2 – b 2 = (a + b) (a – b) Sum or Difference of two cubes a 3 ± b 3 =(a ± b)(a 2 ab+ b 2 ) Basic rules of factoring

Examples x 8 – y 8 3x y 6 x 3 – 27y 3

3—if there are three terms remaining: it is a trinomial make two sets of parenthesis determine which signs to place in them the factors of the first are placed in the first half of each the factors of the last are placed in the last half of each then the middle term is verified using the OI part of FOIL Basic rules of factoring if the sign before the constant is - use a + and a – if the sign before the constant is + use two of the sign before the x term Or memorize Have Use

Examples x 2 + 2x – 83x x x 2 + x – 3

4—if there are four terms remaining: using grouping —look for terms that have a common factor (remember they may need to be rearranged) Basic rules of factoring

Examples Examples: t 3 + 6t 2 – 2t – 12 xy + xz + wy + wz

Alternate method Examples: 20x 2 – 3x – 2 2 ∙ 20 = 40 factors of 40 that subtract to give you three Rewrite: 20x 2 – 3x – 2 As 20x 2 + 5x – 8x – 2 Use grouping 5x(4x + 1) – 2(4x + 1) Since they share (4x + 1) Factor that out of each (4x +1)(5x - 2)

SUMMARIZATION

HOMEWORK PAGE(S): NUMBERS: 4, 8, 12, 16, 50, 54, 62 PAGE(S): NUMBERS: 3 – 66 multiples of three, 72, 78

5-6 General Factoring Strategies EQ: How do you factor when all the types are mixed together?

Basic rules of Factoring a Polynomial Make sure the terms are in order x 2, x, constant Look for a common factor Are there two terms three terms four terms

CLASSWORK 5-6 Factoring: A General Strategy PAGE(S): 231 NUMBERS: 2 – 38 even

5-7 Solving Equations by Factoring EQ: How do you solve equations using factoring?

Activation: Multiply the following: -3∙0=-3∙1=-3∙-1= 2∙0= 2∙1= 2∙-1= What do you notice is true? Solve: 2x – 4 = 0 What does this really mean?

Basic rules of Solving a Polynomial Make sure the terms are in order x 2, x, constant and Make sure all terms are on the same side (i.e. it =0) Look for a common factor Are there two terms three terms four terms Set each factor equal to zero and solve

Example X 2 + 3x – 28 = 012x 2 + x = 6

Example -15x + 2x = 0x 2 = 36

SUMMARIZATION 1. 2.

HOMEWORK PAGE(S): 233 NUMBERS: 2 – 30 even, 36, 43

5-8 Using Equations EQ: How do you translate word problems into equations that can be solved by factoring?

Activation: Five more than a number is 12. Find the number.

Example Four times the square of a number is 21 more than eight times the number. What is the number? Number = x 4x 2 = x

Example The length of the top of a table is 5 ft greater than the width. Find the length and width if the area is 84 ft 2 ? The item compared to is x A = L * W 84 = (x + 5)(x) 84 = x 2 + 5x 0 = x 2 + 5x – 84 0 = (x + 12)(x – 7) 0 = x + 12 or 0 = x – 7 x = -12 or x = 7 Width = x Length = x + 5

Example The sum of the squares of two consecutive even integers is 164. What are the integers? 1st integer = x 2ns integer= x + 2 x 2 + (x+2) 2 = 164 x 2 + x 2 + 4x + 4 = 164 2x 2 + 4x – 160 = 0 2 (x 2 + 2x – 80) = 0 2( x + 10) ( x – 8) = 0

SUMMARIZATION 1. 2.

HOMEWORK PAGE(S): 235 – 236 NUMBERS: 2 – 16 even

Review PAGE(S): 241 NUMBERS: evens