§ 5.5 Factoring Special Forms

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.5 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, Intermediate Algebra, 5e – Slide #3 Section 5.5 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, Intermediate Algebra, 5e – Slide #4 Section 5.5 The Difference of Two SquaresEXAMPLE SOLUTION Factor: 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, Intermediate Algebra, 5e – Slide #5 Section 5.5 The Difference of Two SquaresEXAMPLE SOLUTION Factor: 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, Intermediate Algebra, 5e – Slide #6 Section 5.5 The Difference of Two SquaresEXAMPLE SOLUTION Factor completely: 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, Intermediate Algebra, 5e – Slide #7 Section 5.5 The Difference of Two Squares The factors of are the sum and difference of the expressions being squared CONTINUED Thus,

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.5 Factoring CompletelyEXAMPLE SOLUTION Factor completely: 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, Intermediate Algebra, 5e – Slide #9 Section 5.5 Factoring Special Forms Factoring Perfect Square Trinomials Let A and B be real numbers, variables, or algebraic expressions.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.5 Factoring Perfect Square TrinomialsEXAMPLE SOLUTION Factor: 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, Intermediate Algebra, 5e – Slide #11 Section 5.5 Grouping & Difference of Two SquaresEXAMPLE SOLUTION Factor: 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, Intermediate Algebra, 5e – Slide #12 Section 5.5 Grouping & Difference of Two Squares Factor the difference of two squares. The factors are the sum and difference of the expressions being squared. Simplify CONTINUED Thus,

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.5 The Sum & Difference of Two Cubes Factoring the Sum & Difference of Two Cubes 1)Factoring the Sum of Two Cubes: Same Signs Opposite Signs 2) Factoring the Difference of Two Cubes: Same Signs Opposite Signs

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.5 The Sum & Difference of Two CubesEXAMPLE SOLUTION Factor: Rewrite as the Sum of Two Cubes Factor the Sum of Two Cubes Simplify Thus,

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 5.5 The Sum & Difference of Two CubesEXAMPLE SOLUTION Factor: Rewrite as the Difference of Two Cubes Factor the Difference of Two Cubes Simplify Thus,