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 5u (1/3)m2 -s2t3

Binomial

A polynomial that has two terms

2x + 3 4x – 3y3xy – 14 613 + 39z

Trinomial

A polynomial that has three terms

2x2 – 3x + 1 14 + 32z – 3xmn – m2 + n2

Polynomial

Expressions with several terms that follow patterns.

4x3 + 3x2 + 15x + 23b2 – 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 + 5x2 – 4x + 5What 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 – 54x2 – x + 3x2 – 5 + x2

11-2

Multiply by a Monomial

Examples

(5a)(-3b)3v2(v2 + v + 1)12(a2 + 3ab2 – 3b3 – 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 100

25 = 5 x 5100 = 2 x 2 x 5 x 5GCF = 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 =

GREATEST COMMON FACTOR

vw + wx = w(v + x)

Factoring Polynomials

21x2 – 35y2

=

Factoring Polynomials

13e – 39ef =

Factoring Polynomials

5m + 35 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

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

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

F.O.I.L. Method

F – multiply First termsO – multiply Outer termsI – multiply Inner termsL – multiply Last termsAdd all terms to get product

Example: (2a – b)(3a + 5b)

F – 2a · 3aO – 2a · 5bI – (-b) ▪ 3aL - (-b) ▪ 5b

Example: (x + 6)(x +4)

F – x ▪ xO – x ▪ 4I – 6 ▪ xL – 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

4x3 + 4x2y2 + xy + y3

Group (4x3 + 4x2y2) and factorGroup (xy + y3) and factor4x2(x +y2) + y(x + y2)Answer: (x +y2) (4x2 + y)

Example

2x3 - 2x2y - 3xy2 + 3y3+ xz2 – yz2

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

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

(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

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) = -18Factors: (-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) = 70Factors: (-7)(-10) add to -1714x2 -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) = -12Factors: (-12)(1) add to -113x2 -12x + 1x - 4(3x2 – 12x) + (1x -4)3x(x-4) + 1(1x -4)(x -4)(3x + 1)

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