Factoring Trinomials
Factor: 4wx2 - 64wx + 256w
Solution
Step 1 Factor out the GCF of the terms of the polynomial.
| Factor out 4w. |
4wx2 - 64wx + 256w
= 4w(x2 - 16x + 64) |
Now we will try to factor the trinomial, x
2 - 16x + 64.
Step 2 Count the number of terms and look for factoring patterns.
There are three terms in x2 - 16x + 64.
• The first term, x2, is a perfect square, (x)2.
• The third term, 64, is a perfect square, (8)2.
• The middle term, -16x, is the opposite of
twice the product of x and 8.
Therefore, x2 - 16x + 64 has the form a2 - 2ab + b2 where a is x and b is 8.
| |
a2 - 2ab + b2 |
= (a - b)(a - b) |
| In the pattern, substitute x
for a and 8 for b. |
(x)2 - 2(x)(8) + (8)2 |
= (x - 8)(x - 8) |
| Step 3 Factor completely. (x - 8)(x - 8) cannot be factored further.
But don’t forget the
original factor 4w that
we factored out in Step 1. |
4wx2 - 64wx + 256w |
= 4w(x - 8)(x - 8) |
We multiply to check the factorization.
| Is Is
Is |
4w(x - 8)(x - 8)
4w(x2 - 16x + 64)
4wx2 - 64wx + 256w |
= 4wx2 - 64wx + 256w ? = 4wx2 - 64wx + 256w ?
= 4wx2 - 64wx + 256w ? Yes |
Example 2
Factor: wx2 + 5x2 - 9w - 45
Solution
Step 1 Factor out the GCF of the terms of the polynomial.
The GCF is 1.
Step 2 Count the number of terms and look for factoring patterns.
| There are four terms, so try
factoring by grouping.
|
= wx2 + 5x2 - 9w - 45
|
| Group the first pair of terms
and factor out x2. |
= (wx2 + 5x2) + (-9w - 45)
= x2(w + 5) + (-9w - 45) |
| Group the second pair of terms.
Factor out -9 (rather than +9) to
obtain w + 5, the same binomial
as in the first grouping.
Factor out (w + 5). |
= x2(w + 5) -9(w + 5) = (w + 5)(x2
-9) |
| Step 3 Factor completely.
w + 5 cannot be factored further.
x2 - 9 is a difference of two squares, (x)2 - (3)2.
So we factor x2 - 9 using the pattern for a difference of two squares. |
= (w + 5)(x + 3)(x - 3) |
The result is: wx2 + 5x2 - 9w - 45 = (w + 5)(x +
3)(x - 3).
We multiply to check the factorization.
| Is Is
Is |
(w + 5)(x + 3)(x - 3)
(w + 5)(x2
-9)
wx2 + 5x2 - 9w - 45 |
= wx2 + 5x2 - 9w - 45 ? = wx2 + 5x2 - 9w - 45
?
= wx2 + 5x2 - 9w - 45 ? Yes |
Is (w 5)(x 3)(x 3)¬ wx2 5x2 9w 45 ?
Is (w 5)(x2 9) wx2 5x2 9w 45 ?
Is wx2 9w 5x2 45 wx2 5x2 9w 45 ? Yes