Chapter 11 Polynomials

Add & Subtract 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)m2 -s2t3

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

Trinomial A polynomial that has three terms 2x2 – 3x + 1 14 + 32z – 3x mn – m2 + n2

Polynomial Expressions with several terms that follow patterns. 4x3 + 3x2 + 15x + 2 3b2 – 2b + 4

Coefficient The constant (or numerical) factor in a monomial 3m2 coefficient = 3 u coefficient = 1 -s2t3 coefficient = -1

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

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

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

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

11-2 Multiply by a Monomial

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

Divide and Find Factors 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 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 21x2 – 35y2 =

Factoring Polynomials 13e – 39ef =

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

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

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

Dividing Polynomials by Monomials 2x + 6x2 2x

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

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

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

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

Find Binomial Factors in a Polynomial 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 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)

Example 2x3 - 2x2y - 3xy2 + 3y3+ xz2 – yz2 Group (2x3 - 2x2y2 ) and factor Group (- 3xy2 + 3y3) and factor Group (xz2 – yz2) and factor Answer:

Special Factoring Patterns 11-6 Special Factoring Patterns

11-6 Difference of Squares (a + b)(a – b)= a2 - b2 (x + 5) (x – 5) = x2 - 25

11-6 Squares of Binomials (a + b)2 = a2 + 2ab + b2 Also known as Perfect square trinomials

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

11-7 Factor Trinomials

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

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

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

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

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

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

More on Factoring Trinomials 11-9 More on Factoring Trinomials

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

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)

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)

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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