Factoring Polynomials

Factoring Polynomials

By Juliann Davison

What is a factor?

The numbers that you multiply together to get an answer

• The factors of 12 are 1, 2, 3, 4, 6, and 12

• The factors of 9x² are 1, 3, 9, x, and x²​

Multiple Choice

What are the factors of 36?

• 1, 2, 3, 4, 6, 9, 12, 18, and 36

• 6

• 1, 2, 3, 4, 5, and 6

• 36 is prime and has no factors.

Multiple Choice

What are the factors of 8a?

• 2 and 4

• 1, 2, 4, 8, and a

• you cannot factor 8a.

• 8 and a

1x36

2x18

3x12

4x9

6x6​

Because 36=:

1, 2, 3, 4, 6, 9, 12, 18, and 36

The factors of 36 are:

What is a factor

1 x 8a

2 x 4a

4a x 2

2a x 4

1a x 8​

Because 8a=:

1, 2, 4, 8, and a

The factors of 8a are:

What is a factor

What is Greatest Common Factor

Greatest= Biggest

Common= ​The same between different things

Factor= Numbers that you multiply to get ​an answer

​

**It is the biggest number​ that is a factor of all terms in an expression

​

12x²y= 2(2)(3)(x)(x)(y)

20y= 2(2)(5)(y)

The greatest common factor between 12x²y and 20y is 2(2)(y)= 4y​

Of 12x²y + 20y

4x= 2(2)(x)

2= 2

The greatest common factor between 4x and 2 IS 2​

​

Of 4x + 2?

What is the GCF (Greatest Common Factor)?

Multiple Choice

What is the greatest common factor of 14x³ - 21x?

• 7x²

• 7x³

• 14x³

• 7x

Multiple Choice

What is the greatest common factor of 6x²y² + 12x²y - 8xy² + 10y?

• 6x²y²

• 2xy

• 2y

• 2x²y

Multiple Choice

What is equivalent to 14x³ - 21x?

• 7x

• 7x(x²-1)

• 7x(x³-3x)

• 7x(2x²-3)

Multiple Choice

Which is equivalent to 6x²y² + 12x²y - 8xy² + 10y?

• 6x²y²

• 2xy(3xy+6y-4x+5x²)

• 2y(3x²y+6x²-4xy+5)

• 2x²y(3y+12-8x+5x²y)

GCF Factoring

GCF(Leftovers)

• When factoring, you always want to factor out the GCF first, if there is one.

• Find the GCF, place it outside the ​parentheses.

• Divide every term by the GCF, and the answer goes inside the parentheses.​

Multiple Choice

Factor 18x² - 12x + 24

• 6(3x² - 2x + 4)

• 18(x² - 2/3x + 4/3

• 2(9x² - 6x + 12)

• This can't be factored.

Factoring into 2 Binomials

(x + 2)(x + 3) = x² + 5x + 6, therefore x² + 5x + 6 = (x + 2)(x + 3)

So you can ask yourself, what adds to 7, but multiplies to 12?

The answer: 3 and 4

So x²+7x+12 factors to (x+3)(x+4)​ or (x+4)(x+3)

​

(x+3)(x+4)= x²+7x+12

You know that if you multiply two binomials, the two numbers, m and n, add to give you the middle term and multiply to give you the last term.

(x+m)(x+n)= x² + (m+n)x + mn

Factoring Quadratics

Multiple Choice

What adds to 16 and multiplies to 60?

• 6 and 10

• 2 and 8

• 4 and 15

• 4 and 12

Multiple Choice

Factor x² + 16x + 60

• (x+2)(x+8)

• (x+4)(x+10)

• (x+4)(x+12)

• (x+4)(x+12)

Multiple Choice

Factor x² - 3x - 28

• Not factorable

• (x-3)(x-28)

• (x+7)(x-4)

• (x-7)(x+4)

x² + 3x + 2

= (x+1)(x+2)

​

​

​

Then factor the rest

4x² + 12x + 8

= 4(​x² + 3x + 2)

First, factor out the GCF -->

You can use GCF and factoring together!

So, 4x² + 12x + 8 = 4(x+1)(x+2)

Multiple Choice

Factor the GCF, then factor the rest: 3x² - 18x + 24

• (3x-4)(x-2)

• 3(x - 4)(x - 2)

• 3(x -3)(x - 8)

• Can't be factored

Difference of Two Squares

Some polynomials can be factored easily using certain methods, one of which is Difference of Two Squares

When two things are subtracted

Difference

A value created by squaring another value

EX: ​ 1= 1(1)

4= 2(2)

x² = x(x)

4x²= 2x(2x)​

A perfect square

Difference of Two Squares

Multiple Choice

Is this a difference of two squares: x² - 16

• Yes, because x²=x(x) and 16= 4(4)

• No because x² and 16 are not BOTH perfect squares

• No, because this isn't a difference

Multiple Choice

Is this a difference of two squares: 4x² + 9

• Yes, because

  4x²=2x(2x) and 9= 3(3)

• No because 4x² and 9 are not BOTH perfect squares

• No, because this isn't a difference

Multiple Choice

Is this a difference of two squares: 2x² - 25

• Yes, because

  2x²=1x(1x) and 25= 5(5)

• No because 2x² and 25 are not BOTH perfect squares

• No, because this isn't a difference

1) x² = x(x) 25= 5(5)

2) (x PLUS 5)(​x MINUS 5)

​

Answer: (x+5)(x-5)​

EX: x² - 25

1) Square root both terms

2) Create two factors:

(Square root of first PLUS square root of second​)

(Square root of first MINUS square root of second)​

Steps

Difference of Two Squares

Notice how the 5 and -5 add to zero, which is why there is no x term

Multiple Choice

Factor: x² - 100

• (x+50)(x-50)

• (x+10)(x-10)

• (x+10)(x+10)

• (x-50)(x-50)

Multiple Choice

Factor: 9x² - 49

• (3x + 7)(3x - 7)

• (x+7)(x-7)

• 9(x+7)(x-7)

• Can't be factored

Multiple Choice

Factor: 18x² - 50

• Can't be factored

• 18(x+25)(x-25)

• 2(x+5)(x-5)

• 2(3x + 5)(3x - 5)

Putting It Together

• Always look for a GCF to factor out FIRST!!!

• If there are two terms, look for Difference of Two Squares

• If there are three terms, try to factor by finding the numbers that add to the middle term and multiply to the end term.

• If there is a coefficient on x², you can still factor. You will learn that next lesson.​​

Let's Practice!!!

Multiple Choice

Factor: x² + 4x + 3

• (x+1)(x+3)

• (x-1)(x+4)

• (x-1)(x-3)

• Not Factorable

Multiple Choice

Factor: x² - 7x + 10

• (x-2)(x-5)

• (x+10)(x+1)

• (x+2)(x+5)

• (x-3)(x-4)

Multiple Choice

Factor: 3x² + 15x + 12

• 3(x+1)(x+4)

• 3(x+4)(x+3)

• (3x+4)(x+3)

• Not Factorable

Multiple Choice

Factor: x² - 81

• (x+9)(x-9)

• x(x-81)

• (x-9)(x-9)

• Not Factorable

Multiple Choice

Factor: 2x² - 32

• 2(x+4)(x-4)

• (2x - 16)(x + 2)

• 2(x-8)(x+4)

• Not Factorable