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Chapter 11 Polynomials

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**Add & Subtract Polynomials**

11-1 Add & Subtract Polynomials

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Monomial A constant, a variable, or a product of a constant and one or more variables u (1/3)m s2t3

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**Binomial A polynomial that has two terms 2x + 3 4x – 3y**

3xy – z

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**Trinomial A polynomial that has three terms 2x2 – 3x + 1 14 + 32z – 3x**

mn – m2 + n2

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**Polynomial Expressions with several terms that follow patterns.**

4x3 + 3x2 + 15x + 2 3b2 – 2b + 4

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**Coefficient The constant (or numerical) factor in a monomial**

3m coefficient = 3 u coefficient = 1 -s2t coefficient = -1

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Like Terms Terms that are identical or that differ only in their coefficients Are 2x and 2y similar? Are -3x2 and 2x2 similar?

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Examples x2 + (-4)x + 5 x2 – 4x + 5 What are the terms? x2, -4x, and 5

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**Simplified Polynomial**

A polynomial in which no two terms are similar. The terms are usually arranged in order of decreasing degree of one of the variables

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**Are they Simplified? 2x2 – 5 + 4x + x2 3x + 4x – 5**

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11-2 Multiply by a Monomial

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Examples (5a)(-3b) 3v2(v2 + v + 1) 12(a2 + 3ab2 – 3b3 – 10)

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**Divide and Find Factors**

11-3 Divide and Find Factors

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**GREATEST COMMON FACTOR**

The greatest integer that is a factor of all the given integers.

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2,3,5,7,11,13,17,19,23,29 Prime number - is an integer greater than 1 that has no positive integral factor other than itself and 1.

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**GREATEST COMMON FACTOR**

Find the GCF of 25 and 100 25 = 5 x 5 100 = 2 x 2 x 5 x 5 GCF = 5 x 5 = 25

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**GREATEST COMMON FACTOR**

Find the GCF of 12 and 36 12 = 36 = GCF =

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**GREATEST COMMON FACTOR**

Find the GCF of 14,49 and 56 14 = 49 = 56 = GCF =

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**Factoring Polynomials**

vw + wx = w(v + x)

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**Factoring Polynomials**

21x2 – 35y2 =

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**Factoring Polynomials**

13e – 39ef =

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**Dividing Polynomials by Monomials**

5 = 5(m+ 7)÷5 = m + 7

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**Dividing Polynomials by Monomials**

7x + 14 7 = 7x + 14 = x + 2

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**Dividing Polynomials by Monomials**

6a + 8b 2 = 2(a +4b) ÷ 2 = a + 2b

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**Dividing Polynomials by Monomials**

2x + 6x2 2x

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**Multiply Two Binomials**

11-4 Multiply Two Binomials

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**Multiplying Binomials**

When multiplying two binomials both terms of each binomial must be multiplied by the other two terms

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**Multiplying binomials**

Using the F.O.I.L method helps you remember the steps when multiplying

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**F.O.I.L. Method F – multiply First terms O – multiply Outer terms**

I – multiply Inner terms L – multiply Last terms Add all terms to get product

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**Example: (2a – b)(3a + 5b) F – 2a · 3a O – 2a · 5b I – (-b) ▪ 3a**

L - (-b) ▪ 5b

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Example: (x + 6)(x +4) F – x ▪ x O – x ▪ 4 I – 6 ▪ x L – 6 ▪ 4

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**Find Binomial Factors in a Polynomial**

11-5 Find Binomial Factors in a Polynomial

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Procedure Group the terms in the polynomial as pairs that share a common monomial factor Extract the monomial factor from each pair

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Procedure If the binomials that remain for each pair are identical, write this as a binomial factor of the whole expression The monomials you extracted create a second polynomial. This is the paired factor for the original expression

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**Example 4x3 + 4x2y2 + xy + y3 Group (4x3 + 4x2y2) and factor**

Group (xy + y3) and factor 4x2(x +y2) + y(x + y2) Answer: (x +y2) (4x2 + y)

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**Example 2x3 - 2x2y - 3xy2 + 3y3+ xz2 – yz2**

Group (2x3 - 2x2y2 ) and factor Group (- 3xy2 + 3y3) and factor Group (xz2 – yz2) and factor Answer:

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**Special Factoring Patterns**

11-6 Special Factoring Patterns

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**11-6 Difference of Squares**

(a + b)(a – b)= a2 - b2 (x + 5) (x – 5) = x2 - 25

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**11-6 Squares of Binomials (a + b)2 = a2 + 2ab + b2**

Also known as Perfect square trinomials

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Examples (x + 3)2 = ? (y - 2)2 = ? (s + 6)2 = ?

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11-7 Factor Trinomials

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**Factoring Pattern for x2 + bx + c, c positive**

x2 + 8x + 15 = (x + 3) (x + 5) Middle term is the sum of 3 and 5 Last term is the product of 3 and 5

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**Example y2 + 14y + 40 = (y + 10) (y + 4)**

Middle term is the sum of 10 and 4 Last term is the product of 10 and 4

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**Example y2 – 11y + 18 = (y - 2) (y - 9)**

Middle term is the sum of -2 and -9 Last term is the product of -2 and -9

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**Factoring Pattern for x2 + bx + c, c negative**

x2 - x - 20 = (x + 4) (x - 5) Middle term is the sum of 4 and -5 Last term is the product of 4 and - 5

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**Example y2 + 6y - 40 = (y + 10) (y - 4)**

Middle term is the sum of 10 and -4 Last term is the product of 10 and - 4

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**Example y2 – 7y - 18 = (y + 2) (y - 9)**

Middle term is the sum of 2 and -9 Last term is the product of 2 and -9

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**More on Factoring Trinomials**

11-9 More on Factoring Trinomials

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**11-9 Factoring Pattern for ax2 + bx + c**

Multiply a(c) = ac List the factors of ac Identify the factors that add to b Rewrite problem and factor by grouping

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**Example 2x2 + 7x – 9 List factors: (-2)(9) = -18**

Factors: (-2)(9) add to 7 (2x2 -2x) + (9x – 9) 2x(x -1) + 9(x – 1) (x-1)(2x +9)

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**Example 14x2 - 17x + 5 List factors: (14)(5) = 70**

Factors: (-7)(-10) add to -17 14x2 -7x – 10x + 5 (14x2 – 7x) + (-10x +5) 7x(2x-1)- 5(2x -1) (7x -5)(2x – 1)

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**Example 3x2 - 11x - 4 List factors: (-12)(1) = -12**

Factors: (-12)(1) add to -11 3x2 -12x + 1x - 4 (3x2 – 12x) + (1x -4) 3x(x-4) + 1(1x -4) (x -4)(3x + 1)

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