1. Algebra 1
Section 10.4

2. Special Patterns
A perfect square is an integer such as 36 or 121 which can be expressed as the square of another integer.
The product of a pair of conjugates is a difference of two squares.

3. Factoring the Difference of Squares
Factoring a difference of squares binomial produces a product of conjugates.
x2 – y2 = (x + y)(x – y)

4. Example 1
Factor x2 – 49.
x2 – 72
(x + 7)(x – 7)

5. Example 2
Factor 5 – 125x2.
Begin by factoring out the common factor of 5.
5(1 – 25x2)
5[12 – (5x)2]
5(1 + 5x)(1 – 5x)

6. Example 3
Factor 36x6z2 – 64y2.
Begin by factoring out the common factor of 4.
4(9x6z2 – 16y2)
4[(3x3z)2 – (4y)2]
4(3x3z + 4y)(3x3z – 4y)

7. Special Patterns
Any binomial that is a sum of squares (a2 + b2) cannot be factored using real numbers.
Earlier, we studied:
(x + y)2 = x2 + 2xy + y2
(x – y)2 = x2 – 2xy + y2

8. Special Patterns (x + y)2 = x2 + 2xy + y2 (x – y)2 = x2 – 2xy + y2
The first and last terms are positive perfect squares.
The middle term is twice the product of the square roots of the first and last terms.

9. Factoring Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2

10. Example 4
Factor x2 + 10x + 25.
The first and last terms are perfect squares. The middle term is twice the product of their square roots.
x2 + 2(x)(5) + 52
= (x + 5)2

11. Example 5
Factor 12x2 – 36x + 27.
First, factor out the common factor of 3.
3(4x2 – 12x + 9)
3[(2x)2 – 2(2x)(3) + 32]
3(2x – 3)2

12. Example 6 Factor x6y4 – 8x3y2z + 16z2.
The first and last terms are perfect squares. The middle term is the opposite of twice the product of their square roots.
(x3y2)2 – 2(x3y2)(4z) + (4z)2
= (x3y2 – 4z)2

13. Homework: pp