Chapter 11 Polynomials

11-1 Add & Subtract Polynomials

Monomial A constant, a variable, or a product of a constant and one or more variables -7 5u (1/3)m 2 -s 2 t 3

Binomial l A polynomial that has two terms 2x + 3 4x – 3y 3xy – z

Trinomial l A polynomial that has three terms 2x 2 – 3x z – 3x mn – m 2 + n 2

Polynomial Expressions with several terms that follow patterns. 4x 3 + 3x x + 2 3b 2 – 2b + 4

Coefficient l The constant (or numerical) factor in a monomial l 3m 2 coefficient = 3 l u coefficient = 1 l -s 2 t 3 coefficient = -1

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

Examples l x 2 + (-4)x + 5 l x 2 – 4x + 5 l What are the terms? l x 2, -4x, and 5

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

Are they Simplified? l 2x 2 – 5 + 4x + x 2 l 3x + 4x – 5 l 4x 2 – x + 3x 2 – 5 + x 2

11-2 Multiply by a Monomial

Examples l (5a)(-3b) l 3v 2 (v 2 + v + 1) l 12a(a 2 + 3ab 2 – 3b 3 – 10)

11-3 Divide and Find Factors

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

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

Find the GCF of 25 and = 5 x = 2 x 2 x 5 x 5 GCF = 5 x 5 = 25 GREATEST COMMON FACTOR

Find the GCF of 12 and = 36 = GCF =

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

vw + wx = w(v + x) Factoring Polynomials

21x 2 – 35y 2 = Factoring Polynomials

13e – 39ef = Factoring Polynomials

5m = 5(m+ 7)÷5 = m + 7 Dividing Polynomials by Monomials

7x = 7x = x + 2 Dividing Polynomials by Monomials

6a + 8b 2 = 2(a +4b) ÷ 2 = a + 2b Dividing Polynomials by Monomials

2x + 6x 2 2x Dividing Polynomials by Monomials

11-4 Multiply Two Binomials

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

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

F.O.I.L. Method l F – multiply First terms l O – multiply Outer terms l I – multiply Inner terms l L – multiply Last terms l Add all terms to get product

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

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

11-5 Find Binomial Factors in a Polynomial

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

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

Example 4x 3 + 4x 2 y 2 + xy + y 3 Group (4x 3 + 4x 2 y 2 ) and factor Group (xy + y 3 ) and factor 4x 2 (x +y 2 ) + y(x + y 2 ) Answer: (x +y 2 ) (4x 2 + y)

Example 2x 3 - 2x 2 y - 3xy 2 + 3y 3 + xz 2 – yz 2 Group (2x 3 - 2x 2 y 2 ) and factor Group (- 3xy 2 + 3y 3 ) and factor Group (xz 2 – yz 2 ) and factor Answer:

11-6 Special Factoring Patterns

11-6 Difference of Squares (a + b)(a – b)= a 2 - b 2 (x + 5) (x – 5) = x

11-6 Squares of Binomials (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 Also known as Perfect square trinomials

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

11-7 Factor Trinomials

Factoring Pattern for x 2 + bx + c, c positive x 2 + 8x + 15 = l Find factors of 15 that add to 8 l Replace 8 with the added factors l Factor by Grouping

Example y y + 40 =

Example y 2 – 11y + 18 =

Factoring Pattern for x 2 + bx + c, c negative x 2 - x - 20 =

Example y 2 + 6y - 40 =

Example y 2 – 7y - 18 =

11-9 More on Factoring Trinomials

11-9 Factoring Pattern for ax 2 + bx + c Multiply a(c) = ac List the factors of ac Identify the factors that add to b Rewrite problem and factor by grouping

Example 2x 2 + 7x – 9 List factors: (-2)(9) = -18 Factors: (-2)(9) add to 7 (2x 2 -2x) + (9x – 9) 2x(x -1) + 9(x – 1) (x-1)(2x +9)

Example 14x x + 5 List factors: (14)(5) = 70 Factors: (-7)(-10) add to x 2 -7x – 10x + 5 (14x 2 – 7x) + (-10x +5) 7x(2x-1)- 5(2x -1) (7x -5)(2x – 1)

Example 3x x - 4 List factors: (-12)(1) = -12 Factors: (-12)(1) add to -11 3x 2 -12x + 1x - 4 (3x 2 – 12x) + (1x -4) 3x(x-4) + 1(1x -4) (x -4)(3x + 1)

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