Factoring Quadratic Expressions

• Trinomial –

• Binomial –

• Monomial –

Consisting of three terms (Ex: 5x3 – 9x2 + 3)

Consisting of 2 terms (Ex: 2x6 + 2x)

Consisting of one term (Ex: x2)

Specific Expressions

An expression in x that can be written in the standard form:

ax2 + bx +c

Where a, b, and c are any number except a ≠ 0.

Quadratic Expression

Factoring

The process of rewriting a mathematical expression involving a sum to a product. It is the opposite of distributing.

Example:

2 10 24 2 12x x x x SUM PRODUCT

Factor

If x2 + 8x + 15 = ( x + 3 )( x + 5 )

then

x + 3 and x + 5 are

called factors of x2 + 8x + 15

(Remember that 3 and 4 are factors of 12 since 3.4=12)

10x

4x2 -6x

-15

Finding the Dimensions of a Generic Rectangle

Mr. Wells’ Way to find the product for a generic rectangle:

First, find the POSITIVE Greatest Common

Factor of two terms in the bottom row.

2x

Second, find missing WHOLE

NUMBER dimensions

on the individual

boxes.2x -3

5

Make sure to Check

Lastly, write the answer as a Product:

2 5 2 3x x

The product of one diagonal always equals the product of the other diagonal.

10x

4x2 -6x

-15 10x . -6x = -60x2

4x2 . -15 = -60x2

A Pattern with Generic Rectangles

Example:

7x

(2x2)(6) 12x2

3x

4x

+6

2x2

ax2 is always in the bottom

left corner

Factoring with the Box and Diamond22 7 6x x

___ 4x 3x

Because of our pattern, the missing

boxes need to multiply to:

The missing boxes also have to add up

to bx in the sum

2x

x 2

3

( 2x + 3 )( x + 2 )

GCF

Factor: c is always in the top right

corner

Dia

mo

nd P

robl

em

Fill in the results from the diamond and find the dimensions of the box:

Write the expression as a product:

(3x2)(-10)

-30x2-1015x

-2x 3x2ax2c

ax2

Factoring Example 23 13 10x x

___ bx

13x

-2x 15x

c Product

Sum

x

3x -2

5

( x + 5 )( 3x – 2 )

GCF

Factor:

33x

-35x

(15x2)(-77)

-1155x2-77

15x2ax2c

ax2

Factoring: Different Order22 15 77x x

___ bx

-2x

-35x 33x

c Product

Sum

5x

3x -7

11

( 5x + 11 )( 3x – 7 )

GCF

Factor:215 2 77x x

Rewrite in Standard Form: ax2 + bx + c

3x

3x

(x2)(9)

9x29

x2ax2c

ax2

Factoring: Perfect Square2 6 9x x

___ bx

6x

3x 3x

c Product

Sum

x

x 3

3

( x + 3 )2

GCF

Factor:

( x + 3 )( x + 3 )

6x

-6x

(9x2)(-4)

-36x2-4

9x2ax2c

ax2

Factoring: Missing Terms29 4x

___ bx

0

-6x 6x

c Product

Sum

3x

3x -2

2

( 3x + 2 )( 3x – 2 )

GCF

Factor:29 0 4x x

Factoring: Which Expression is correct?

24 10 6x x Factor:

If you use the box and diamond, the following products are possible:

3 4 2

2 6 2 1

x x

and

x x

Which is the best possible answer?

Notice that every term is divisible by 2

÷2x2

Factoring: Factoring Completely210 25 15x x

210 25 15x x 5

22x 35( x + 3 )( 2x – 1 ) 25 2 5 3x x

Factor:

5x

6x

-x

(2x2)(-3)

-6x2-3

2x2ax2c

ax2___ bx

5x

-x 6x

c Product

Sum

x

2x -1

3

GCF

Reverse Box to factor out the GCF Ignore the GCF and factor the quadratic

Don’t forget the GCF

Factoring: Factoring Completely3 23 6 45x x x

3 23 6 45x x x 3x

2x 153x(x + 3)(x – 5) 23 2 15x x x

Factor:

2x

3x

-5x

(x2)(-15)

-15x2-15

x2ax2c

ax2___ bx

-2x

-5x 3x

c Product

Sum

x

x -5

3

GCF

Reverse Box to factor out the GCF Ignore the GCF and factor the quadratic

Don’t forget the GCF

Factoring: Forgot to Factor a GCF

5 1 4x x

5 20x 5x 4

When factoring the above expression a student came up with the following answer. Is it factored

completely?

No, it is not factored completely because one of the factors still

has a GCF bigger than 1.

2 25 15 20x x Factor:

1 5 20x x Factor out the

GCF with a reverse box

Substitute the result:

1 5 4x x NOTE: I do not recommend relying on this. It can be used IF you forget to check for a GCF.

Factoring: Just Factoring a GCF24 20x x

4 5x x

24 20x x4xx 5

Factor:

Reverse Box to factor out the GCF

There is no longer a quadratic, it is not possible to factor

anymore. There is not always more factoring

after the GCF.

Factoring: Ensuring “a” is Positive2 13 42x x

2 13 42x x 1

2x 42-( x + 6 )( x + 7 ) 2 13 42x x

Factor:

13x

6x

7x

(x2)(42)

42x242

x2ax2c

ax2___ bx

13x

6x 7x

c Product

Sum

x

x 7

6

GCF

Reverse Box to factor out the negative Ignore the GCF and factor the quadratic

Don’t forget the GCF

When the x2 term is negative, it is difficult to factor.