D AY 137 – F ACTORING S PECIAL C ASES
E XAMPLE 1 Examine each expression. Is the expression a perfect-square trinomial? a) 𝑦 2 + 8𝑦 + 16 b) 3𝑦 2 + 16𝑦 + 16 c) 4𝑦 2 − 3𝑦𝑧 − 𝑧 2 d) 4𝑏 2 + 12𝑏𝑐 + 9𝑐 2 e) 9𝑧 2 + 6𝑦𝑧 + 𝑦 2 f) 16𝑦 2 + 12𝑦𝑧 + 𝑧 2
A NSWER a) Yes; 𝑦 + 4 2 b) No; 3𝑦 2 is not a perfect square. c) No; 4𝑦 2 and 𝑧 2 are perfect squares, but the sign in front of 𝑧 2 is negative, not positive. Also the middle term does not equal 2 2𝑦 (𝑧) . d) Yes; 2𝑏 + 3𝑐 2 e) Yes; 3𝑧 + 𝑦 2 f) No, 16𝑦 2 and 𝑧 2 are perfect squares, but the middle term is not 2(4𝑦)(𝑧) . The perfect-square trinomial pattern can be used to factor expressions in the form of 𝑏 2 + 2𝑏𝑐 + 𝑐 2 or 𝑏 2 − 2𝑏𝑐 + 𝑐 2 .
F ACTORING A P ERFECT -S QUARE T RINOMIAL For all numbers 𝑏 and 𝑐 . 𝑏 2 + 2𝑏𝑐 + 𝑐 2 = 𝑏 + 𝑐 𝑏 + 𝑐 = 𝑏 + 𝑐 2 , and 𝑏 2 − 2𝑏𝑐 + 𝑐 2 = 𝑏 − 𝑐 𝑏 − 𝑐 = 𝑏 − 𝑐 2 .
E XAMPLE 2 Factor each expression. a) 𝑦 2 − 10𝑦 + 25 b) 9𝑡 2 + 24𝑡 + 16 c) 64𝑏 2 − 16𝑏𝑐 + 𝑐 2 d) 49𝑧 4 + 14𝑧 2 + 1
A NSWER a) 𝑦 − 5 2 b) 3𝑦 + 4 2 c) 8𝑏 − 𝑐 2 d) 7𝑧 2 + 1 2 To check, substitute a number for the variable and evaluate. The are of a square is 𝑦 2 + 6𝑦 + 9 square units. If the length of one side is 5 units, find the value of 𝑦 by factoring.
D IFFERENCE OF T WO S QUARES When you multiply 𝑏 + 𝑐 𝑏 − 𝑐 , the product is 𝑏 2 − 𝑐 2 . This product is called the difference of two squares. ( a b )( a b ) a ( a b ) b ( a b ) 2 2 a ab ab b 2 2 a b
E XAMPLE 3 Examine each expression. Is the expression a difference of two squares? Explain a) 4𝑏 2 − 25 b) 9𝑦 2 − 15 c) 𝑐 2 + 49 d) 𝑑 2 − 4𝑒 2 e) 𝑏 3 − 9
A NSWER a) Yes; 2𝑏 2 − 5 2 b) No; 15 is not a perfect a difference; it is a sum. c) No; 𝑐 2 + 49 is not a difference; it is a sum. d) Yes; 𝑑 2 − 2𝑒 2 e) No; 𝑏 3 is not a perfect square.
A NSWER The difference-of-two-square pattern can be used to factor expressions in the form 𝑏 2 𝑐 2 For Example, 4𝑑 2 − 81𝑒 2 fits the pattern The first term is a perfect square, 2𝑑 2 The second term is a perfect square, 9𝑒 2 The terms are subtracted. Thus, 4𝑑 2 − 81𝑒 2 = 2𝑑 + 9𝑒 2𝑑 − 9𝑒
F ACTORING A D IFFERENCE OF T WO S QUARES For all number 𝑏 and 𝑐 , 𝑏 2 − 𝑐 2 = (𝑏 + 𝑐)(𝑏 − 𝑐) .
E XAMPLE 4 Factor each expression. a) 𝑦 2 − 4 b) 36𝑏 2 − 49𝑐 2 c) 16𝑦 2 − 25 d) 𝑛 4 − 𝑜 4
A NSWER a) 𝑦 + 2 𝑦 − 2 b) (6𝑏 + 7𝑐)(6𝑏 − 7𝑐) c) (4𝑦 + 5)(4𝑦 − 5) d) 𝑛 2 + 𝑜 2 𝑛 2 − 𝑜 2 = (𝑛 2 + 𝑜 2 )(𝑛 + 𝑜)(𝑛 − 𝑜)
E XAMPLE 5 Find each product by using the difference of two squares. a) 31 ∙ 29 b) 17 ∙ 13 c) 34 ∙ 26
A NSWER a) Think of 31 ∙ 29 as (30 + 1)(30 − 1) . The product is 30 2 − 1 2 = 900 − 1 = 899. b) Think 17 ∙ 13 as (15 + 2)(15 − 2) . The product is 15 2 − 2 2 = 255 − 4 = 211. c) Think of 34 ∙ 26 as (30 + 4)(30 − 4) . The product is 30 2 − 4 2 = 900 − 16 = 884.
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