#### Transcript Example 2: Factoring a Polynomial

### 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