Polynomials: Factoring - XYZ Custom Plus

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764Chapter 12 <strong>Polynomials</strong>: <strong>Factoring</strong>NoteIf you think the sum oftwo squares x 2 + y 2factors, you shouldtry it. Write down the factors youthink it has, and then multiplythem using the foil method. Youwon’t get x 2 + y 2 .The factor (x2 − y 2 ) is itself the difference of two squares and therefore can be fac-tored again. The factor (x2 + y 2 ) is the sum of two squares and cannot be factoredagain. The complete solution is this:Bx 4 − y 4 = (x2 ) 2 − ( y 2 ) 2= (x2 − y 2 )(x2 + y 2 )= (x+ y)(x− y)(x2 + y 2 )Perfect Square Trinomials4. Factor 4x2 + 12x + 9.NoteAs we have indicatedbefore, perfectsquare trinomialslike the ones in Examples 4 and 5can be factored by the methodsdeveloped in previous sections.Recognizing that they factor tobinomial squares simply saves timein factoring.ExAmplE 4SOlutiOnFactor 25x2 − 60x+ 36.Although this trinomial can be factored by the method we used inSection 6.3, we notice that the first and last terms are the perfect squares (5x) 2and (6) 2 . Before going through the method for factoring trinomials by listing allpossible factors, we can check to see if 25x 2 − 60x+ 36 factors to (5x − 6) 2 . Weneed only multiply to check:(5x− 6) 2 = (5x− 6)(5x− 6)= 25x2 − 30x− 30x+ 36= 25x2 − 60x+ 36The trinomial 25x2 − 60x+ 36 factors to (5x− 6)(5x− 6) = (5x− 6) 2 .5. Factor 6x2 + 24x+ 24.ExAmplE 5Factor 5x2 + 30x+ 45.SOlutiOn We begin by factoring out the greatest common factor, which is 5.Then we notice that the trinomial that remains is a perfect square trinomial:5x2 + 30x+ 45 = 5(x2 + 6x+ 9)= 5(x+ 3) 26. Factor (x− 4) 2 − 36.ExAmplE 6 Factor (x− 3) 2 − 25.SOlutiOnThis example has the form a 2 − b 2 , where a is x − 3 and b is 5. Wefactor it according to the formula for the difference of two squares:(x− 3) 2 − 25 = (x− 3) 2 − 5 2 Write 25 as 5 2= [(x− 3) − 5][(x− 3) + 5] Factor= (x− 8)(x+ 2) SimplifyNotice in this example we could have expanded (x− 3) 2 , subtracted 25, and thenfactored to obtain the same result:(x− 3) 2 − 25 = x 2 − 6x+ 9 − 25 Expand (x − 3) 2= x 2 − 6x− 16 Simplify= (x− 8)(x+ 2) FactorAnswers3. (a 2 + b 2 )(a + b)(a − b)4. (2x + 3) 2 5. 6(x+ 2) 26. (x− 10)(x+ 2)Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Describe how you factor the difference of two squares.2. What is a perfect square trinomial?3. How do you know when you’ve factored completely?4. Describe how you would factor 25x 2 − 60x + 36.
12.4 Problem Set765Problem Set 12.4A Factor the following. [Examples 1–3]1. x 2 − 9 2. x 2 − 25 3. a 2 − 364. a 2 − 64 5. x 2 − 49 6. x 2 − 1217. 4a 2 − 16 8. 4a 2 + 16 9. 9x 2 + 2510. 16x 2 − 36 11. 25x 2 − 169 12. x 2 − y 213. 9a 2 − 16b 2 14. 49a 2 − 25b 2 15. 9 − m 216. 16 − m 2 17. 25 − 4x 2 18. 36 − 49y 219. 2x 2 − 18 20. 3x 2 − 27 21. 32a 2 − 12822. 3a 3 − 48a 23. 8x 2 y − 18y 24. 50a 2 b − 72b25. a 4 − b 4 26. a 4 − 16 27. 16m 4 − 8128. 81 − m 4 29. 3x 3 y − 75xy 3 30. 2xy 3 − 8x 3 y
Page 1 and 2: Polynomials: Factoring12Introductio
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Page 45 and 46: 12.7 Problem Set78358. (x + 7) 2 =
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