### Factoring a Polynomial

Example 1: Factoring a Polynomial Completely factor

x

3 + 2

x

2 – 11

x

– 12 Use the graph or table to find at least one real root. x = -4 is a real root because it is an x-intercept.

Since x = -4 is a root, (x + 4) is a factor of the original cubic equation.

Now use polynomial division to “factor out” the (x + 4).

y

x

3  2

x

2  11

x

 12

Example 1: Factoring a Polynomial Completely factor

x

3 + 2

x

2 – 11

x

– 12

x

2 -2

x

-3 Now we can rewrite the 

x

 4   cubic:

x

2  2

x

 3 

x x

3 -2

x

2 -3

x

This quadratic can be factored using old techniques: (x + 1)(x – 3) + 4 4

x

2 -8

x

-12 Since the graph of the cubic had more than one real root, this may be able to be

x

3 + 2

x

2 – 11

x

Thus, the completely factored form is: 

x

 – 12 4 

x

 1  factored more.

x

 3 

### Let’s try another example.

Example 2: Factoring a Polynomial Completely factor

x

4 –

x

3 + 4

x

– 16 Use the graph or table to find at least one real root. x = -2 is a real root because it is an x-intercept.

Since x = -2 is a root, (x + 2) is a factor of the original degree 4 equation.

Now use polynomial division to “factor out” the (x + 2).

y

x

4 4

x

 16

Example 2: Factoring a Polynomial

x

Completely factor

x

4 –

x

3 + 4

x

– 16

x

3

x

4 -3

x

2 -3

x

3 6

x

6

x

2 -8 -8

x

Now we can rewrite the degree 4 equation: 

x

 2  

x

3  3

x

2  6

x

 8  Let’s check the graph of this cubic to see if it has a real root.

+ 2 2

x

3 -6

x

2 12

x

-16

x

4 –

x

3 + 0

x

2 + 4

x

– 16 Make sure to include all powers of

x

Since the graph of the degree 4 equation had more than one real root, this may be able to be factored more.

Example 2: Factoring a Polynomial Completely factor

x

4 Current Factored form:  –

x x

3  + 4

x

2  

x

3 – 16  3

x

2  6

x

 8  Use the graph or table of the cubic in the factored form to find at least one real root.

y

x

3  3

x

2  6

x

 8 x = 2 is a real root because it is an x-intercept.

Since x = 2 is a root, (x – 2) is a factor of the cubic in the factored form.

Now use polynomial division to “factor out” the (x – 2) of the cubic in the factored form.

Example 2: Factoring a Polynomial Completely factor

x

4 Current Factored form:  –

x x

3  + 4

x

2  

x

3 – 16  3

x

2  6

x

 8 

x

2 -

x

4 Now we can rewrite the 

x

current factored form as:  2 

x

 2  

x

2 4 

x x

3 -

x

2 4

x

This quadratic can NOT be factored using old techniques (No x-intercepts).

– 2 -2

x

2 2

x

-8 Since the graph of the cubic had only one real root, this may NOT

x

3 – 3

x

2 + 6

x

Thus, the completely factored form is: 

x

 – 8 2 

x

be able to be factored more.

 2  

x

2 4 