Chapter 5

Factoring Polynomials

5-1 Factoring Integers

Factors - integers that are multiplied together to produce a product.

4 x5 = 20

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

Prime factorization of 3636 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3 = 22 x 32

PRIME FACTORIZATION

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

GREATEST COMMON FACTOR

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

5-2 Dividing Monomials

Property of Quotients

If a, b, c and d are real numbers, then

ac =a • cbd b • d

If b, c and d are real numbers, then

bc =cbd d

Simplifying Fractions

If a is a nonzero real number and m and n are positive integers, and m > n, then am = am-n

an

Rule of Exponents for Division

If a is a nonzero real number and m and n are positive integers, and

n > m; then am = 1 an an-m

Rule of Exponents for Division

If a is a nonzero real number and m and n are positive integers, and

m = n; then am = 1 an

Rule of Exponents for Division

The greatest common factor of two or more monomials is the common factor with the greatest coefficient and the greatest degree in each variable.

GREATEST COMMON FACTOR

Find the GCF of 25x4y and 50x2y5

GCF = 25x2y

GREATEST COMMON FACTOR

5-3 Monomial Factors

of Polynomials

Dividing a Polynomial by a Monomial

Divide each term of the polynomial by the monomial and add the results.

5m + 35 = m + 7 5 7x2 + 14x = x + 2 7x

Dividing Polynomials by Monomials

To factor:1.Find the GCF2.Divide each term by the GCF

3.Write the product

Factoring a Polynomial

Examples

5x2 + 10x

4x5 – 6x3 + 14x

8a2bc2 – 12ab2c2

5-4 Multiplying Binomials Mentally

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

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

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) ▪ 5b6a2 + 10ab – 3ab – 5b2 6a2 + 7ab – 5b2

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

F – x ▪ xO – x ▪ 4 I – 6 ▪ xL – 6 ▪ 4

x2 + 4x + 6x + 24x2 + 10x + 24

Section 5-5 Difference of Two

Squares

Multiplying

(x + 3) (x - 3) = ?

(y - 2)(y + 2) = ?

(s + 6)(s – 6) = ?

Factoring Pattern

a2 – b2 =(a –b) (a + b)

FACTOR

x2 - 49 = ?

16 – y2 = ?

81t2 – 25x6 = ?

5-6 Squares of Binomials

Examples - Multiply

(x + 3)2 = ?

(y - 2)2 = ?

(s + 6)2 = ?

Factoring Patterns

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

• Also known as Perfect square trinomials

Examples – Factor

1. 4x2 + 20x + 252. 64u2 + 72uv + 81v2

3. 9m2 – 12m + 44. 25y2 + 5y + 1

5-7 Factoring Pattern for x2 + bx + c, c positive

Example

x2 + 8x + 15Middle term is the sum of 3 and 5

Last term is the product of 3 and 5

Example

y2 + 14y + 40

Middle term is the sum of 10 and 4

Last term is the product of 10 and 4

Example

y2 – 11y + 18

Middle term is the sum of -2 and -9

Last term is the product of -2 and -9

Factor

1. m2 – 3m + 52. k2 + 9k + 203. y2 – 9y + 8

5-8 Factoring Pattern for

x2 + bx + c, c negative

x2 - x - 20

Middle term is the sum of 4 and -5

Last term is the product of 4 and -5

Example

y2 + 6y - 40

Middle term is the sum of 10 and -4

Last term is the product of 10 and -4

Example

y2 – 7y - 18

Middle term is the sum of 2 and -9

Last term is the product of 2 and -9

Factor

1. x2 – 4kx – 12k2

2. p2 – 32p – 333. a2 + 3ab – 18b2

5-9 Factoring Pattern for ax2 + bx + c

• List the factors of ax2

• List the factors of c• Test the possibilities to see which produces the correct middle term

Examples

2x2 + 7x – 914x2 - 17x + 510 + 11x – 6x2

5a2 – ab – 22b2

5 -10 Factor by Grouping

Factor each polynomial by grouping terms that have a common factor

Then factor out the common factor and write the polynomial as a product of two factors

Examples

xy – xz – 3y + 3z3xy – 4 – 6x + 2y xy + 3y + 2x + 6 ab – 2b + ac – 2c 9p2 – t2 – 4ts – 4s2

5 -11 Using Several Methods of Factoring

A polynomial is factored completely when it is expressed as the product of a monomial and one or more prime polynomials.

Guidelines for Factoring Completely

Factor out the greatest monomial factor first

Factor the remaining polynomial

Guidelines for Factoring Completely

Make sure that each binomial or trinomial factor is prime.

Example - Factor

-4n4 + 40n3 – 100n2

5a3b2 + 3a4b – 2a2b3

a2bc - 4bc + a2b - 4b

5 -12 Solving Equations by Factoring

Zero-Product PropertyFor all real numbers a and b:

ab = 0 if and only if a = 0 or b = 0

Examples

1. (x + 2) (x – 5) = 02. 5n(n – 3)(n – 4) = 03. 2x2 + 5x = 124. 18y3 + 8y + 24y2 = 0

5 -13 Using Factoring to Solve Word Problems

Suppose Mike bought 36 feet of wire to make a rectangular pen for his pet. If he wants the area to be 80 ft2, what are the dimensions he could use?

Solution

Let x= Length, then Width = (36 – 2x)/2 = 18 – x80 = 18x – x2

x2 – 18x + 80 = 0(x – 10) (x-8) = 0

{8, 10}

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