5.4 Factoring Special Products
Learning Objectives
By the end of this section, you should be able to:
- Recognize and factor perfect-square trinomials.
- Recognize and factor the difference of squares.
- Verify special-product patterns by expansion.
- Identify prime expressions that fit neither pattern.
Perfect-Square Trinomials
A perfect-square trinomial results when a binomial is multiplied by itself:
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
How to Recognize the Pattern
- The first and last terms are perfect squares.
- The middle term is twice the product of their square roots.
- The sign of the middle term matches the sign in the binomial.
Example 5.4.1
Factor: x² + 10x + 25
Solution
Square roots are x and 5.
2(x)(5) = 10x → matches the middle term.
(x + 5)²
Example 5.4.2
Factor: y² − 12y + 36
Solution
Square roots are y and 6.
−2(y)(6) = −12y → matches the middle term.
(y − 6)²
Example 5.4.3
Factor: a² + 14a + 45
Solution
45 is not a perfect square.
This is not a perfect-square trinomial.
Factor normally:
(a + 5)(a + 9)
Try It
Difference of Squares
The difference of squares pattern occurs when one square is subtracted from another:
a² − b² = (a + b)(a − b)
Important Notes:
- The expression must have two terms only.
- The sign between the terms must be minus.
- Sum of squares does not factor over the real numbers.
Example 5.4.4
Factor: x² − 16
Solution
Square roots are x and 4.
(x + 4)(x − 4)
Example 5.4.5
Factor: 9a² − 25b²
Solution
Square roots are 3a and 5b.
(3a + 5b)(3a − 5b)
Example 5.4.6
Factor: 8x² − 50
Solution
First factor out the GCF:
2(4x² − 25)
Now apply difference of squares:
4x² − 25 = (2x + 5)(2x − 5)
2(2x + 5)(2x − 5)
Example 5.4.7
Factor: x² + 9
Solution
There is no middle term, but the sign is positive.
Prime (cannot be factored over the real numbers)
Try It
Recognizing and Verifying Special Products
To verify a special product:
- Take square roots of the first and last terms.
- Double their product and compare to the middle term.
- If the middle term is missing and the sign is minus, check for difference of squares.
Example 5.4.8
Factor each expression.a) 4x² − 9
Solution
(2x + 3)(2x − 3)
b) y² + 14y + 49
Solution
(y + 7)²
c) x² + 36
Solution
Prime
d) 16a² − 81b²
Solution
(4a + 9b)(4a − 9b)
Try It
Key Takeaways
- Perfect-square trinomials:
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b² - Difference of squares:
a² − b² = (a + b)(a − b) - Always factor out a GCF first.
- Sum of squares is prime over the real numbers.
- Verify any factorization by expanding.
