1. § 5.5
Factoring Special Forms

2. Special Polynomials
In this section we will consider some polynomials that have special forms
that make it easy for us to see how they factor. You may look at a polynomial and say, “Oh, that’s just a difference of squares” or “I think we have a sum of cubes here.” When you have a special polynomial, in particular one that is a difference of two squares, a perfect square polynomial, or a sum or difference of cubes, you will have a factoring formula memorized and will know how to proceed.
That’s why these polynomials are “special”. They may just become our best
friends among the polynomials.….
Blitzer, Algebra for College Students, 6e – Slide #2 Section 5.5

3. The Difference of Two Squares
If A and B are real numbers, variables, or algebraic expressions, then
In words: The difference of the squares of two terms, factors as the product of a sum and a difference of those terms.
Blitzer, Algebra for College Students, 6e – Slide #3 Section 5.5

4. The Difference of Two Squares
EXAMPLE
Factor:
SOLUTION
We must express each term as the square of some monomial. Then we use the formula for factoring
Express as the difference of two squares
Factor using the Difference of Two Squares method
Blitzer, Algebra for College Students, 6e – Slide #4 Section 5.5

5. The Difference of Two Squares
EXAMPLE
Factor:
SOLUTION
The GCF of the two terms of the polynomial is 6. We begin by factoring out 6.
Factor the GCF out of both terms
Factor using the Difference of Two Squares method
Blitzer, Algebra for College Students, 6e – Slide #5 Section 5.5

6. The Difference of Two Squares
EXAMPLE
Factor completely:
SOLUTION
Express as the difference of two squares
The factors are the sum and difference of the expressions being squared
The factor is the difference of two squares and can be factored
Blitzer, Algebra for College Students, 6e – Slide #6 Section 5.5

7. The Difference of Two Squares
CONTINUED
The factors of are the sum and difference of the expressions being squared
Thus,
Blitzer, Algebra for College Students, 6e – Slide #7 Section 5.5

8. Factoring Completely Factor completely:
EXAMPLE
Factor completely:
SOLUTION
Group terms with common factors
Factor out the common factor from each group
Factor out x + 3 from both terms
Factor as the difference of two squares
Blitzer, Algebra for College Students, 6e – Slide #8 Section 5.5

9. Factoring Perfect Square Trinomials
Factoring Special Forms
Factoring Perfect Square Trinomials
Let A and B be real numbers, variables, or algebraic expressions.
Blitzer, Algebra for College Students, 6e – Slide #9 Section 5.5

10. Factoring Perfect Square Trinomials
EXAMPLE
Factor:
SOLUTION
We suspect that is a perfect square trinomial because The middle term can be expressed as twice the product of 4x and -5y.
Express in form
Factor
Blitzer, Algebra for College Students, 6e – Slide #10 Section 5.5

11. Grouping & Difference of Two Squares
EXAMPLE
Factor:
SOLUTION
Group as minus a perfect square trinomial to obtain a difference of two squares
Factor the perfect square trinomial
Rewrite as the difference of two squares
Blitzer, Algebra for College Students, 6e – Slide #11 Section 5.5

12. Grouping & Difference of Two Squares
CONTINUED
Factor the difference of two squares. The factors are the sum and difference of the expressions being squared.
Simplify
Thus,
Blitzer, Algebra for College Students, 6e – Slide #12 Section 5.5

13. Factoring the Sum & Difference of Two Cubes
Factoring the Sum of Two Cubes:
Same Signs Opposite Signs
2) Factoring the Difference of Two Cubes:
Blitzer, Algebra for College Students, 6e – Slide #13 Section 5.5

14. The Sum & Difference of Two Cubes
EXAMPLE
Factor:
SOLUTION
Rewrite as the Sum of Two Cubes
Factor the Sum of Two Cubes
Simplify
Thus,
Blitzer, Algebra for College Students, 6e – Slide #14 Section 5.5

15. The Sum & Difference of Two Cubes
EXAMPLE
Factor:
SOLUTION
Rewrite as the Difference of Two Cubes
Factor the Difference of Two Cubes
Simplify
Thus,
Blitzer, Algebra for College Students, 6e – Slide #15 Section 5.5