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§ 7.4

§ 7.4. Factoring Special Forms. Special Polynomials. In this section we will consider some polynomials that have special forms

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§ 7.4

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  1. § 7.4 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 from the beginning 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, Introductory Algebra, 5e – Slide #2 Section 7.4

  3. The Difference of Two Squares Blitzer, Introductory Algebra, 5e – Slide #3 Section 7.4

  4. The Difference of Perfect Squares EXAMPLE • Factor x2 - 25 x2 - 25 = x2 – 52 = (x + 5) (x - 5) • Factor 9x2 - 4y2 9x2 - 4y2 = (3x)2 – (2y)2 = (3x + 2y) (3x - 2y) A2 – B2 (A+B) (A-B) A2 – B2 (A+B) (A-B) Blitzer, Introductory Algebra, 5e – Slide #4 Section 7.4

  5. The Difference of Squares EXAMPLE • Factor 3x2 - 75. There is a common factor here. Remove it first. 3x2 - 75 = 3(x2 – 25) = 3(x2 – 52) = 3(x + 5) (x - 5) Factor the Difference of Squares, A2 – B2 (A+B) (A-B) Factor out the GCF. Blitzer, Introductory Algebra, 5e – Slide #5 Section 7.4

  6. 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, Introductory Algebra, 5e – Slide #6 Section 7.4

  7. 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, Introductory Algebra, 5e – Slide #7 Section 7.4

  8. 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, Introductory Algebra, 5e – Slide #8 Section 7.4

  9. The Difference of Two Squares CONTINUED The factors of are the sum and difference of the expressions being squared Thus, Blitzer, Introductory Algebra, 5e – Slide #9 Section 7.4

  10. Factoring 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, Introductory Algebra, 5e – Slide #10 Section 7.4

  11. Factoring Special Forms Blitzer, Introductory Algebra, 5e – Slide #11 Section 7.4

  12. Perfect Square Trinomials Let A and B be real numbers, variables or algebraic expressions • A2 +2AB + B2 = (A + B)2 • A2 -2AB + B2 = (A - B)2 If the first and last terms are each perfect squares and the middle term is twice the product of the square roots of those two terms – then you have a perfect square trinomial. Blitzer, Introductory Algebra, 5e – Slide #12 Section 7.4

  13. Perfect Square Trinomials EXAMPLE • Factor x2 + 10x + 25 x2 + 10x + 25 = (x)2 + 2(5x) + (5)2 = (x + 5)2 • Factor x2 - 12x + 36 x2 - 12x + 36 = (x)2 – 2(6x) + (6)2 = (x - 6)2 A2 +2AB + B2 (A+B)2 (A-B)2 A2 –2AB+B2 Blitzer, Introductory Algebra, 5e – Slide #13 Section 7.4

  14. 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, Introductory Algebra, 5e – Slide #14 Section 7.4

  15. 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, Introductory Algebra, 5e – Slide #15 Section 7.4

  16. 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, Introductory Algebra, 5e – Slide #16 Section 7.4

  17. The Sum or Difference of Two Cubes Blitzer, Introductory Algebra, 5e – Slide #17 Section 7.4

  18. Sum or Difference of Two Cubes Formulas to memorize: Let A and B be real numbers, variables or algebraic expressions. • Factoring a Sum of Perfect Cubes. A3 + B3 = (A + B)(A2 – AB + B2) • Factoring a Difference of Perfect Cubes. A3 - B3 = (A - B)(A2 + AB + B2) Note that each of these expressions factors into a binomial times a trinomial. A good way to remember the signs is SOP (same, opposite, plus), where “same” means the same sign as in the expression you are factoring. Blitzer, Introductory Algebra, 5e – Slide #18 Section 7.4

  19. Sum of Cubes EXAMPLE Factor x3 + 8. In this example, we first note that we do have the sum of two perfect cubes. The first term is the cube of x, and the second the cube of 2. (A = x, B = 2) x3 + 23 = (x + 2)(x2 – 2x + 22) = (x + 2)(x2 – 2x + 4) A3 + B3 = (A + B)(A2 – AB + B2) Blitzer, Introductory Algebra, 5e – Slide #19 Section 7.4

  20. Difference of Cubes EXAMPLE Factor 8y3 – 27. In this example, we note that we have the difference of two perfect cubes. The first term is the cube of 2y, and the second term, 27, is the cube of 3. (A = 2y, B = 3) Then, the polynomial factors as: (2y)3 - 33 = (2y - 3)((2y)2 + 2y(3) + 32) = (2y - 3)(4y2 +6y + 9) A3 - B3 = (A - B)(A2 + AB + B2) Blitzer, Introductory Algebra, 5e – Slide #20 Section 7.4

  21. The Sum or Difference of Two Cubes EXAMPLE Factor: SOLUTION Rewrite as the Sum of Two Cubes Factor the Sum of Two Cubes Simplify Thus, Blitzer, Introductory Algebra, 5e – Slide #21 Section 7.4

  22. The Sum or Difference of Two Cubes EXAMPLE Factor: SOLUTION Rewrite as the Difference of Two Cubes Factor the Difference of Two Cubes Simplify Thus, Blitzer, Introductory Algebra, 5e – Slide #22 Section 7.4

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