The rules for factoring perfect squares depend on whether you're dealing with a perfect square trinomial or a difference of squares.
Factoring Perfect Square Trinomials
Perfect square trinomials follow specific patterns that make them easy to factor. The general forms are:
- a2 + 2ab + b2 = (a + b)2
- a2 - 2ab + b2 = (a - b)2
According to the provided reference, to factor a perfect square trinomial, you should:
- Verify the form: Ensure the trinomial is in the form a2 + 2ab + b2 or a2 - 2ab + b2.
- Check the middle term: Confirm that the middle term is twice the product of the square roots of the first and last terms (2 a b).
- Check the sign: Note the sign of the middle term. This will determine whether you use (a + b)2 or (a - b)2.
Example:
Factor x2 + 6x + 9
- Verify the form: This looks like a perfect square trinomial.
- Check the middle term:
- √x2 = x (This is 'a')
- √9 = 3 (This is 'b')
- 2 x 3 = 6x (Matches the middle term!)
- Check the sign: The middle term is positive, so we use the (a + b)2 form.
Therefore, x2 + 6x + 9 = (x + 3)2
Factoring the Difference of Squares
Another common type of perfect square factoring involves the difference of squares.
The general form is:
- a2 - b2 = (a + b)(a - b)
Example:
Factor x2 - 16
- Identify 'a' and 'b':
- √x2 = x (This is 'a')
- √16 = 4 (This is 'b')
Therefore, x2 - 16 = (x + 4)(x - 4)
Summary Table
| Type of Expression | Form | Factored Form | Example |
|---|---|---|---|
| Perfect Square Trinomial (Positive) | a2 + 2ab + b2 | (a + b)2 | x2 + 4x + 4 = (x + 2)2 |
| Perfect Square Trinomial (Negative) | a2 - 2ab + b2 | (a - b)2 | x2 - 6x + 9 = (x - 3)2 |
| Difference of Squares | a2 - b2 | (a + b)(a - b) | x2 - 25 = (x + 5)(x - 5) |
