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2. 2 6.4 Factoring Polynomials with Special Forms

3. 3 What You Will Learn  Factor the difference of two squares  Factor a polynomial completely  Identify and factor perfect square trinomials  Factor the sum or difference of two cubes

4. 4 Difference of Two Squares

5. 5 One of the easiest special polynomial forms to recognize and to factor is the form a 2 – b 2. It is called a difference of two squares and it factors according to the following pattern.

6. 6 Difference of Two Squares To recognize perfect square terms, look for coefficients that are squares of integers and for variables raised to even powers. Here are some examples. Original Difference Polynomial of Squares Factored Form x 2 – 1 (x) 2 – (1) 2 (x + 1)(x – 1) 4x 2 – 9 (2x) 2 – (3) 2 (2x + 3)(2x – 3)

7. 7 Example 1 – Factoring the Difference of Two Square Factor each polynomial. a. x 2 – 36 b. x 2 – c. 81x 2 – 49 Solution: a. x 2 – 36 = x 2 – 6 2 = (x + 6)(x – 6) Write as difference of two squares. Factored form

8. 8 cont’d b. x 2 – = x 2 – = c. 81x 2 – 49 = (9x) 2 – 7 2 = (9x + 7)(9x – 7) Check your results by using the FOIL Method. Write as difference of two squares. Factored form Write as difference of two squares. Example 1 – Factoring the Difference of Two Square

9. 9 The rule a 2 – b 2 = (a + b)(a – b) applies to polynomials or expressions in which a and b are themselves expressions. Sometimes the difference of two squares can be hidden by the presence of a common monomial factor. Remember that with all factoring techniques, you should first remove any common monomial factors. Difference of Two Squares

10. 10 Example 2 – Removing a Common Monomial Factor First A hammer is dropped from the roof of a building. The height of the hammer is given by the expression –16t 2 + 64, where t is the time in seconds. a.Factor the expression b.How many seconds does it take the hammer to fall to a height of 41 feet? Solution a. –16t 2 + 64= –16(t 2 – 4) = –16(t 2 – 2 2 ) = –16(t + 2)(t – 2) Factor out common monomial factor Write a difference of two squares Factored form

11. 11 Example 2 – Removing a Common Monomial Factor First b. Use a spreadsheet to find the heights of the hammer at 0.1-second intervals of the time t. From the spreadsheet, you can see that the hammer falls to a height of 41 feet in about 1.2 seconds. cont’d

12. 12 Factoring Completely

13. 13 Factoring Completely To factor a polynomial completely, you should always check to see whether the factors obtained might themselves be factorable. That is, can any of the factors be factored? For instance, after factoring the polynomial (x 4 – 1) once as the difference of two squares (x 4 – 1) = (x 2 ) 2 – 1 2 = (x 2 + 1)(x 2 – 1) you can see that the second factor is itself the difference of two squares. Factored form Write as difference of two squares.

14. 14 Factoring Completely So, to factor the polynomial completely, you must continue the factoring process. (x 4 – 1) = (x 2 + 1)(x 2 – 1) = (x 2 + 1)(x + 1)(x – 1) Factor completely. Factor as difference of two squares.

15. 15 Example 3 – Removing a Common Monomial Factor First Factor the polynomial 20x 3 – 5x. Solution: 20x 3 – 5x = 5x(4x 2 – 1) = 5x[(2x) 2 – 1 2 ] = 5x(2x + 1)(2x – 1) Factor out common monomials factor 5x Write as difference of two squares Factored form

16. 16 Example 4 – Factoring Completely Factor the polynomial x 4 – 16 completely. Solution: Recognizing x 4 – 16 as a difference of two squares, you can write x 4 – 16 = (x 2 ) 2 – 4 2 = (x 2 + 4)(x 2 – 4). Note that the second factor (x 2 – 4) is itself a difference of two squares, and so x 4 – 16 = (x 2 + 4)(x 2 – 4) = (x 2 + 4)(x + 2)(x – 2). Write as difference of two squares. Factored form Factor as difference of two squares. Factor completely.

17. 17 Perfect Square Trinomials

18. 18 Perfect Square Trinomial A perfect square trinomial is the square of a binomial. For instance, x 2 + 4x + 4 = (x + 2)(x + 2) = (x + 2) 2 is the square of the binomial (x + 2).

19. 19 Perfect square trinomials come in two forms: one in which the middle term is positive and the other in which the middle term is negative. In both cases, the first and last terms are positive perfect squares. Perfect Square Trinomial

20. 20 Example 5 – Identifying Perfect Square Trinomials Which of the following are perfect square trinomials? a. m 2 – 4m + 4 b. 4x 2 – 2x + 1 c. y 2 + 6y – 9 d. x 2 + x +

21. 21 cont’d Example 5 – Identifying Perfect Square Trinomials a. This polynomial is a perfect square trinomial. It factors as (m – 2) 2. b. This polynomial is not a perfect square trinomial because the middle term is not twice the product of 2x and 1. c. This polynomial is not a perfect square trinomial because the last term, –9, is not positive. d. This polynomial is a perfect square trinomial. It factors as.

22. 22 Example 6 – Factoring a Perfect Square Trinomial Factor the trinomial y 2 – 6y + 9. Solution y 2 – 6y + 9= y 2 – 2(3y) + 9 Recognize the pattern = (y – 3) 2 Write in factored form

23. 23 Example 7 – Factoring a Perfect Square Trinomial Factor the trinomial 16x 2 + 40y + 25. Solution 16x 2 + 40y + 25 = (4x) 2 + 2(4x)(5) + 5 2 Recognize the pattern = (4x + 5) 2 Write in factored form

24. 24 Example 8 – Factoring a Perfect Square Trinomial Factor the trinomial 9x 2 + 24xy + 16y 2. Solution 9x 2 + 24xy + 16y 2 = (3x) 2 + 2(3x)(4y) + (4y) 2 = (3x + 4y) 2

25. 25 Sum or Difference of Two Cubes

26. 26 Perfect Square Trinomial The last type of special factoring presented in this section is the sum or difference of two cubes.

27. 27 Example 9 – Factoring the Sum or Difference of two Cubes, a. Factor the polynomial y 3 + 27. Solution: y 3 + 27 = y 3 + 3 3 = (y + 3) [y 2 – (y)(3) + 3 2 ] = (y + 3)(y 2 – 3y + 9) Write as sum of two cubes. Factored form Simplify.

28. 28 Example 9 – Factoring the Sum or Difference of two Cubes, b. Factor the polynomial 64 – x 3. Solution: 64 – x 3 = y 3 + 3 3 = (4 – x)[4 2 + (4)(x) + x 2 ] = (4 – x)(16 + 4x + x 2 ) Write as sum of two cubes. Factored form Simplify. cont’d

29. 29 Example 9 – Factoring the Sum or Difference of two Cubes, c. Factor the polynomial 2x 3 – 16. Solution: 2x 3 – 16 = 2(x 3 – 8) = 2(x 3 – 2 3 ) = 2(x – 2)[x 2 + (x)(2) + 2 2 ] = 2(x – 2)(x 2 + 2x + 4) Factor out common monomial factor 2 Write as sum of two cubes. Factored form Simplify cont’d