Factoring Trinomials Using Patterns
If we recognize that a given polynomial follows a certain pattern, we can
use that pattern to factor the polynomial.
Pattern —
To Factor a Perfect Square Trinomial
| Perfect Square Trinomial: |
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2 |
Example 1
Factor: 25y2 - 10x2y + x4
Solution
Check the form of 25y2 - 10x2y + x4:
• there are three terms;
• the first term is a perfect square: 25y2 = (5y)2;
• the third term is a perfect square: x4 = (x2)2; and
• the second term is twice the product of the factors being squared:
10x2y = 2(5y)(x2).
| So, the given polynomial is
a perfect square trinomial:
Substitute 5y for a and x2 for b: |
a2 - 2ab + b2
(5y)2 - 2(5y)(x2) + (x2)2 |
= (a + b)2 = (5y - x2)2 |
Thus, the factorization is (5y - x2)2.
You can multiply to check the factorization.
Note:
We can use FOIL to check the
factorization:
| (5y - x2)2 |
= (5y - x2)(5y - x2)
= 25y2 - 5x2y - 5x2y + x4
= 25y2 - 10x2y + x4 |
The factorization checks.
Pattern —
To Factor the Difference of Two Perfect Squares
Difference of Two Perfect Squares: a2 - b2 = (a + b)(a
- b)
Example 2
Factor: 16w2 - 81y6.
Solution
Check the form of 16w2 - 81y6.
• there are two terms;
• the first term is a perfect square: 16w2 = (4w)2;
• the second term is a perfect square: 81y6 = (9y3)2.
| So, the given polynomial is a
difference of two squares:
Substitute 4w for a and 9y3 for b: |
a2 - b2
(4w)2 - (9y3)2 |
= (a + b)(a - b) = (4w + 9y3)(4w - 9y3) |
Thus, the factorization is (4w + 9y3)(4w - 9y3). You can multiply to check the factorization.
Note:
The polynomial 16w2 + 81y6 is not a
difference of two squares because the
terms are added rather than subtracted.
The polynomial 16w2 + 81y6 cannot be
factored using integers.
We can also use patterns to factor the sum or difference of two cubes.
Patterns —
To Factor the Sum or Difference of Two Cubes
Sum of Two Cubes: a3 + b3 = (a + b)(a2 - ab
+ b2)
Difference of Two Cubes: a3 - b3 = (a - b)(a2 + ab
+ b2)
Example 3
Factor: 8y3 - 27x6
Solution
Check the form of 8y3 - 27x6:
• there are two terms;
• the first term is a perfect cube: 8y3 = (2y)3;
• the second term is a perfect cube: 27x6 = (3x2)3.
So, the polynomial is a difference of two cubes.
| Substitute 2y for a and 3x2 for b:
Simplify. |
a3 - b3
(2y)3 - (3x2)3 |
= (a - b) (a2 + ab + b2)
= (2y - 3x2)[(2y)2 + (2y)(3x2) + (3x2)2]
= (2y - 3x2)(4y2 + 6x2y + 9x4)
|
Thus, the factorization is (2y - 3x2)(4y2 + 6x2y
+ 9x4).
You can multiply to check the factorization. |
