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