Factoring Polynomials Factor

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Chapter 13

¡ì 13.1

Factoring

Polynomials

The Greatest Common

Factor

Chapter Sections

Factors

13.1 ¨C The Greatest Common Factor

13.2 ¨C Factoring Trinomials of the Form

x2

Factors (either numbers or polynomials)

When an integer is written as a product of

integers, each of the integers in the product is a

factor of the original number.

When a polynomial is written as a product of

polynomials, each of the polynomials in the

product is a factor of the original polynomial.

Factoring ¨C writing a polynomial as a product of

polynomials.

+ bx + c

13.3 ¨C Factoring Trinomials of the Form ax2 + bx + c

13.4 ¨C Factoring Trinomials of the Form x2 + bx + c

by Grouping

13.5 ¨C Factoring Perfect Square Trinomials and

Difference of Two Squares

13.6 ¨C Solving Quadratic Equations by Factoring

13.7 ¨C Quadratic Equations and Problem Solving

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Greatest Common Factor

Greatest Common Factor

Example

Greatest common factor ¨C largest quantity that is a

factor of all the integers or polynomials involved.

Find the GCF of each list of numbers.

1) 6, 8 and 46

6=2¡¤3

8=2¡¤2¡¤2

46 = 2 ¡¤ 23

So the GCF is 2.

2) 144, 256 and 300

144 = 2 ¡¤ 2 ¡¤ 2 ¡¤ 3 ¡¤ 3

256 = 2 ¡¤ 2 ¡¤ 2 ¡¤ 2 ¡¤ 2 ¡¤ 2 ¡¤ 2 ¡¤ 2

300 = 2 ¡¤ 2 ¡¤ 3 ¡¤ 5 ¡¤ 5

So the GCF is 2 ¡¤ 2 = 4.

Finding the GCF of a List of Integers or Terms

1) Prime factor the numbers.

2) Identify common prime factors.

3) Take the product of all common prime factors.

? If there are no common prime factors, GCF is 1.

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Greatest Common Factor

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Greatest Common Factor

Example

Example

Find the GCF of each list of numbers.

1) 12 and 8

12 = 2 ¡¤ 2 ¡¤ 3

8=2¡¤2¡¤2

So the GCF is 2 ¡¤ 2 = 4.

2) 7 and 20

7=1¡¤7

20 = 2 ¡¤ 2 ¡¤ 5

There are no common prime factors so the

GCF is 1.

Martin-Gay, Developmental Mathematics

Find the GCF of each list of terms.

1) x3 and x7

x3 = x ¡¤ x ¡¤ x

x7 = x ¡¤ x ¡¤ x ¡¤ x ¡¤ x ¡¤ x ¡¤ x

So the GCF is x ¡¤ x ¡¤ x = x3

2) 6x5 and 4x3

6x5 = 2 ¡¤ 3 ¡¤ x ¡¤ x ¡¤ x

4x3 = 2 ¡¤ 2 ¡¤ x ¡¤ x ¡¤ x

So the GCF is 2 ¡¤ x ¡¤ x ¡¤ x = 2x3

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Greatest Common Factor

Factoring out the GCF

Example

Example

Factor out the GCF in each of the following

polynomials.

Find the GCF of the following list of terms.

a3b2, a2b5 and a4b7

a3b2 = a ¡¤ a ¡¤ a ¡¤ b ¡¤ b

a2b5 = a ¡¤ a ¡¤ b ¡¤ b ¡¤ b ¡¤ b ¡¤ b

a4b7 = a ¡¤ a ¡¤ a ¡¤ a ¡¤ b ¡¤ b ¡¤ b ¡¤ b ¡¤ b ¡¤ b ¡¤ b

So the GCF is a ¡¤ a ¡¤ b ¡¤ b = a2b2

1) 6x3 ¨C 9x2 + 12x =

3 ¡¤ x ¡¤ 2 ¡¤ x2 ¨C 3 ¡¤ x ¡¤ 3 ¡¤ x + 3 ¡¤ x ¡¤ 4 =

3x(2x2 ¨C 3x + 4)

2) 14x3y + 7x2y ¨C 7xy =

7 ¡¤ x ¡¤ y ¡¤ 2 ¡¤ x2 + 7 ¡¤ x ¡¤ y ¡¤ x ¨C 7 ¡¤ x ¡¤ y ¡¤ 1 =

7xy(2x2 + x ¨C 1)

Notice that the GCF of terms containing variables will use the

smallest exponent found amongst the individual terms for each

variable.

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Factoring Polynomials

11

Factoring out the GCF

Example

The first step in factoring a polynomial is to

find the GCF of all its terms.

Factor out the GCF in each of the following

polynomials.

Then we write the polynomial as a product by

factoring out the GCF from all the terms.

1) 6(x + 2) ¨C y(x + 2) =

6 ¡¤ (x + 2) ¨C y ¡¤ (x + 2) =

(x + 2)(6 ¨C y)

2) xy(y + 1) ¨C (y + 1) =

xy ¡¤ (y + 1) ¨C 1 ¡¤ (y + 1) =

(y + 1)(xy ¨C 1)

The remaining factors in each term will form a

polynomial.

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Factoring

Factoring Trinomials

Remember that factoring out the GCF from the terms of

a polynomial should always be the first step in factoring

a polynomial.

Recall by using the FOIL method that

F

I

L

x2

(x + 2)(x + 4) = + 4x + 2x + 8

= x2 + 6x + 8

This will usually be followed by additional steps in the

process.

To factor x2 + bx + c into (x + one #)(x + another #),

note that b is the sum of the two numbers and c is the

product of the two numbers.

Example

Factor 90 + 15y2 ¨C 18x ¨C 3xy2.

90 + 15y2 ¨C 18x ¨C 3xy2 = 3(30 + 5y2 ¨C 6x ¨C xy2) =

3(5 ¡¤ 6 + 5 ¡¤ y2 ¨C 6 ¡¤ x ¨C x ¡¤ y2) =

3(5(6 + y2) ¨C x (6 + y2)) =

3(6 + y2)(5 ¨C x)

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So we¡¯ll be looking for 2 numbers whose product is

c and whose sum is b.

Note: there are fewer choices for the product, so

that¡¯s why we start there first.

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Factoring Polynomials

Example

¡ì 13.2

Factoring Trinomials of the

Form x2 + bx + c

Factor the polynomial x2 + 13x + 30.

Since our two numbers must have a product of 30 and a

sum of 13, the two numbers must both be positive.

Positive factors of 30

Sum of Factors

1, 30

31

2, 15

17

3, 10

13

Note, there are other factors, but once we find a pair

that works, we do not have to continue searching.

So x2 + 13x + 30 = (x + 3)(x + 10).

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Factoring Polynomials

Prime Polynomials

Example

Example

Factor the polynomial x2 ¨C 11x + 24.

Since our two numbers must have a product of 24 and a

sum of -11, the two numbers must both be negative.

Negative factors of 24

Sum of Factors

¨C 1, ¨C 24

¨C 25

¨C 2, ¨C 12

¨C 14

¨C 3, ¨C 8

¨C 11

Factor the polynomial x2 ¨C 6x + 10.

Since our two numbers must have a product of 10 and a

sum of ¨C 6, the two numbers will have to both be negative.

Negative factors of 10

Sum of Factors

¨C 1, ¨C 10

¨C 11

¨C 2, ¨C 5

¨C7

Since there is not a factor pair whose sum is ¨C 6,

x2 ¨C 6x +10 is not factorable and we call it a prime

polynomial.

So x2 ¨C 11x + 24 = (x ¨C 3)(x ¨C 8).

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Factoring Polynomials

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Check Your Result!

Example

Factor the polynomial x2 ¨C 2x ¨C 35.

Since our two numbers must have a product of ¨C 35 and a

sum of ¨C 2, the two numbers will have to have different signs.

Factors of ¨C 35

Sum of Factors

¨C 1, 35

34

1, ¨C 35

¨C 34

¨C 5, 7

2

5, ¨C 7

¨C2

You should always check your factoring

results by multiplying the factored polynomial

to verify that it is equal to the original

polynomial.

Many times you can detect computational

errors or errors in the signs of your numbers

by checking your results.

So x2 ¨C 2x ¨C 35 = (x + 5)(x ¨C 7).

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