Chapter 11 Polynomials 11 1 Add Subtract Polynomials

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 5 u 2 (1/3)m 2 3 -s t

Binomial l. A polynomial that has two terms 2 x + 3 3 xy – 14 4 x – 3 y 613 + 39 z

Trinomial l. A polynomial that has three terms 2 2 x – 3 x + 1 14 + 32 z – 3 x 2 2 mn – m + n

Polynomial Expressions with several terms that follow patterns. 4 x 3 + 3 x 2 + 15 x + 2 3 b 2 – 2 b + 4

Coefficient l The constant (or numerical) factor in a monomial l 3 m 2 coefficient = 3 lu 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 2 x and 2 y similar? l Are -3 x 2 and 2 x 2 similar?

Examples 2 lx + (-4)x + 5 2 l x – 4 x + 5 l What are the terms? l x 2, -4 x, 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? 2 l 2 x 2 x – 5 + 4 x + l 3 x + 4 x – 5 l 4 x 2 – x + 3 x 2 – 5 + x 2

11 -2 Multiply by a Monomial

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

11 -3 Divide and Find Factors

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

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.

GREATEST COMMON FACTOR Find the GCF of 25 and 100 25 = 5 x 5 100 = 2 x 5 x 5 GCF = 5 x 5 = 25

GREATEST COMMON FACTOR Find the GCF of 12 and 36 12 = 36 = GCF =

GREATEST COMMON FACTOR Find the GCF of 14, 49 and 56 14 = 49 = 56 = GCF =

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

Factoring Polynomials 2 21 x = – 2 35 y

Factoring Polynomials 13 e – 39 ef =

Dividing Polynomials by Monomials 5 m + 35 5 = 5(m+ 7)÷ 5 = m+7

Dividing Polynomials by Monomials 7 x + 14 7 = 7 x + 14 7 7 = x+2

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

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

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

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

Example 2 x 3 - 2 x 2 y - 3 xy 2 + 3 y 3+ xz 2 – yz 2 Group (2 x 3 - 2 x 2 y 2 ) and factor Group (- 3 xy 2 + 3 y 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)= 2 a - 2 b (x + 5) (x – 5) = x 2 - 25

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

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

11 -7 Factor Trinomials

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

Example y 2 + 14 y + 40 =

Example y 2 – 11 y + 18 =

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

Example y 2 + 6 y - 40 =

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

Example 2 14 x - 17 x + 5 List factors: (14)(5) = 70 Factors: (-7)(-10) add to -17 14 x 2 -7 x – 10 x + 5 2 (14 x – 7 x) + (-10 x +5) 7 x(2 x-1)- 5(2 x -1) (7 x -5)(2 x – 1)

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

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