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HFCC Math Lab Beginning Algebra – 17 FACTORING THE DIFFERENCE OF TWO SQUARES Introductory Terminology Terms Definitions Product An indicated multiplication; for example: 2 ⋅ 3, 4 ⋅ a, 7b, 5t(a + 3) are all products. Factor Any one of the numbers or expressions being multiplied to form a product; for example: in 4 ⋅ a, 4 and a are factors; in 5t(a + 3), 5, t, (a + 3) are factors. Perfect square Any number or expression that can be obtained by multiplying a quantity by itself; for example: 25 is a perfect square since 25 = 5 ⋅ 5 or 52 . 4t 2 is a perfect square since 4t 2 = (2t)(2t) = (2t )2 . Objective I. To identify perfect squares Objective II. To write the difference of two squares as the product of two factors. I. EXAMPLES: Directions: 1. 9x 2 Identify which of the following are perfect squares. Think: What value times itself equals 9x 2 ? (3 x) 2 = 9x 2 , so 9x 2 is a perfect square. 2. 64 Think: What value times itself equals 64? 82 = 64, so 64 is a perfect square. 3. y4 Think: What value times itself equals y 4 ? y 4 = y 2 ⋅ y 2 = ( y 2 )2 . y 4 is a perfect square. 4. y5 Revised 03/09 Think: The power of a perfect square is an even number. y 5 is not a perfect square since it is impossible to square a whole number power and get an odd power. 1 5. y6 y 6 is a perfect square since y 6 = ( y 3 ) 2 . 6. y9 y 9 is not a perfect square by the same reason as #4. 7. 1 1 is a perfect square since 12 = 1. 8. x2 y 4 x 2 y 4 is a perfect square since x 2 y 4 = ( xy 2 ) . 2 II. To factor (write as a product) any expression of the form ( A) − ( B ) , re-write this as the product of a sum times a difference, namely (A + B)(A – B); that is, A2 − B 2 = ( A + B )( A − B ) or ( A − B)( A + B) . 2 2 EXAMPLES: Directions: Factor each expression completely. 1. 64 − t 2 Think: Is 64 a perfect square? Yes, 64 = 82 . Is t 2 a perfect square? Yes, t 2 = (t ) 2 . 64 − t 2 = (8)2 − (t ) 2 . To factor a difference of perfect squares, use the formula above. 64 − t 2 = (8 + t)(8 – t). 2. 9x2 −1 Think: Is 9 x 2 a perfect square? Yes, 9 x 2 = (3 x) 2 . Is 1 a perfect square? Yes, 1 = (1) 2 . 9 x 2 − 1 = (3 x) 2 − (1)2 . To factor a difference of perfect squares, use the formula above. 9 x 2 − 1 = (3x + 1)(3x – 1). 3. 64 + t 2 Think: This is not a difference of two squares and thus cannot be factored as above. 4. y 4 − 25 Can you think this through for yourself? y 4 − 25 = ( y 2 )2 − (5) 2 , so y 4 − 25 = ( y 2 − 5)( y 2 + 5) . 5. x 2 y 4 − 9t 2 Try this for yourself, too. The answer is: x 2 y 4 − 9t 2 = ( xy 2 ) 2 − (3t )2 and so equals the product ( xy 2 − 3t )( xy 2 + 3t ) . 6. 5 x 2 − 45 Think: Are 5 x 2 or 45 perfect squares? Can you factor a common factor from 5 x 2 and 45? Revised 03/09 2 The expression 5 x 2 − 45 , has 5 as a common factor and can be factored as 5( x 2 − 9), now factor x 2 − 9 . So, 5 x 2 − 45 = 5(x - 3)(x + 3). Exercises: Factor completely. 1. p2 − q 2 11. b 2 c 2 − 9d 2 2. y2 − 9 12. 25m2 − 16n 2 3. 9 −t2 13. 9 g 2 − 121 4. 4a 2 − 9b 2 14. 4 x 2 − 16 y 2 5. 16b 2 − 1 15. 81 − 4h 2 6. 25 y 2 − 9 16. x 4 − 16 7. a 2 − 36b 2 17. a4 − 1 8. 49 x 2 − 16 y 2 18. 2t 2 − 32 9. 4 − 9x2 19. 28a 3 − 7at 2 10. 100 − x 2 y 2 20. 3a 2b 2 − 12c 2 Solutions to odd-numbered problems and answers to even-numbered problems: 1. ( p 2 − q2 ) (p + q)(p – q) 2. (y – 3)(y + 3) 3. 9 −t2 32 − t 2 (3 – t)(3 + t) 14. 4(x -2y)(x + 2y) 15. 81 − 4h 2 92 − (2h)2 (9 – 2h)(9 + 2h) (2a + 3b)(2a – 3b) 16. ( x − 2)( x + 2)( x 2 + 4) 4. Revised 03/09 13. 9 g 2 − 121 (3 g )2 − 112 (3g – 11)(3g + 11) 3 5. 16b 2 − 1 (4b)2 − 12 (4b+1)(4b-1) 17. (a 2 − 1)(a 2 + 1) (a − 1)(a + 1)(a 2 + 1) 6. (5y – 3)(5y + 3) 18. 2(t – 4)(t + 4) 19. 28a 3 − 7at 2 7a (4a 2 − t 2 ) 7. a 2 − 36b 2 2 ( a ) − (6b)2 (a + 6b)(a – 6b) 2 7 a ( 2a ) − t 2 8. (7x + 4y)(7x – 4y) 7a(2a – t)(2a + t) 9. 4 − 9x 2 22 − (3 x)2 (2 – 3x)(2 + 3x) 10. (10 – xy)(10 + xy) 11. (bc )2 − (3d ) 2 (bc + 3d)(bc – 3d) 12. (5m – 4n)(5m + 4n) 20. 3(ab - 2c)(ab + 2c) NOTE: You can get additional instruction and practice by going to the following websites. http://regentsprep.org/Regents/math/factor/Lfactps.htm This website provides step by step instruction and exercises. http://algebralab.org/practice/practice.aspx?file=Algebra_Redden4B.x ml This website provides one worksheet with answers. http://www.cbv.ns.ca/gbhigh/pauloneill/Objective17.08.pdf This website provides step by step instruction and exercises. Revised 03/09 4