Factoring a Polynomial
Example 1: Factoring a Polynomial
Completely factor x3 + 2x2 – 11x – 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  2 x  11 x  12
3
2
Example 1: Factoring a Polynomial
Completely factor x3 + 2x2 – 11x – 12
x2
3
-2x
2
-3
x
x
-2x
-3x
+4
4x2
-8x
-12
x3
+ 2x2 – 11x – 12
Thus, the completely
factored form is:
Now we can rewrite the
cubic:
 x  4 x
2
 2 x  3
This quadratic can be factored
using old techniques:
(x + 1)(x – 3)
Since the graph of the cubic
had more than one real
root, this may be able to be
factored more.
 x  4   x  1  x  3 
Let’s try another example.
Example 2: Factoring a Polynomial
Completely factor x4 – x3 + 4x – 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  x  4 x  16
4
3
Example 2: Factoring a Polynomial
Completely factor x4 – x3 + 4x – 16
x3
-3x2 6x
-8
Now we can rewrite the
degree 4 equation:
 x  2  x  3x  6 x  8
3
x
x
4
-3x
3
2
6x
+ 2 2x3 -6x2 12x
-8x
-16
x4 – x3 + 0x2 + 4x – 16
Make sure to include all powers of x
2
Let’s check the graph of this cubic
to see if it has a real root.
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 x4 – x3 + 4x – 16
Current Factored form:  x  2   x  3 x  6 x  8 
Use the graph or table of the
cubic in the factored form to
find at least one real root.
3
2
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.
y  x  3x  6 x  8
3
2
Now use polynomial division to
“factor out” the (x – 2) of the
cubic in the factored form.
Example 2: Factoring a Polynomial
Completely factor x4 – x3 + 4x – 16
Current Factored form:  x  2   x  3 x  6 x  8 
Now we can rewrite the
2
x
-x
4
current factored form as:
3
x
x3
-x2
4x
–2
-2x2
2x
-8
x3
– 3x2 + 6x
Thus, the completely
factored form is:
–8
2
 x  2 x  2 x
2
 x  4
This quadratic can NOT be factored
using old techniques (No x-intercepts).
Since the graph of the cubic had
only one real root, this may NOT
be able to be factored more.
 x  2 x  2 x
2
 x  4