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E XAMPLE 1 Examine each expression. Is the expression a - - PowerPoint PPT Presentation
E XAMPLE 1 Examine each expression. Is the expression a - - PowerPoint PPT Presentation
D AY 137 F ACTORING S PECIAL C ASES E XAMPLE 1 Examine each expression. Is the expression a perfect-square trinomial? a) 2 + 8 + 16 b) 3 2 + 16 + 16 c) 4 2 3 2 d) 4 2 + 12 + 9 2 e)
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ANSWER
a) Yes; 𝑦 + 4 2 b) No; 3𝑦2 is not a perfect square. c) No; 4𝑦2 and 𝑧2 are perfect squares, but the sign in front
- f 𝑧2 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, 16𝑦2 and 𝑧2 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 𝑏2 + 2𝑏𝑐 + 𝑐2 or 𝑏2 − 2𝑏𝑐 + 𝑐2.
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FACTORING A PERFECT-SQUARE TRINOMIAL
For all numbers 𝑏 and 𝑐. 𝑏2 + 2𝑏𝑐 + 𝑐2 = 𝑏 + 𝑐 𝑏 + 𝑐 = 𝑏 + 𝑐 2, and 𝑏2 − 2𝑏𝑐 + 𝑐2 = 𝑏 − 𝑐 𝑏 − 𝑐 = 𝑏 − 𝑐 2.
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EXAMPLE 2
Factor each expression. a) 𝑦2 − 10𝑦 + 25 b) 9𝑡2 + 24𝑡 + 16 c) 64𝑏2 − 16𝑏𝑐 + 𝑐2 d) 49𝑧4 + 14𝑧2 + 1
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ANSWER
a) 𝑦 − 5 2 b) 3𝑦 + 4 2 c) 8𝑏 − 𝑐 2 d) 7𝑧2 + 1 2 To check, substitute a number for the variable and evaluate. The are of a square is 𝑦2 + 6𝑦 + 9 square
- units. If the length of one side is 5 units,
find the value of 𝑦 by factoring.
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DIFFERENCE OF TWO SQUARES
2 2 2 2
) ( ) ( ) )( ( b a b ab ab a b a b b a a b a b a
When you multiply 𝑏 + 𝑐 𝑏 − 𝑐 , the product is 𝑏2 − 𝑐2. This product is called the difference of two squares.
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EXAMPLE 3
Examine each expression. Is the expression a difference of two squares? Explain a) 4𝑏2 − 25 b) 9𝑦2 − 15 c) 𝑐2 + 49 d) 𝑑2 − 4𝑒2 e) 𝑏3 − 9
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ANSWER
a) Yes; 2𝑏 2 − 52 b) No; 15 is not a perfect a difference; it is a sum. c) No; 𝑐2 + 49 is not a difference; it is a sum. d) Yes; 𝑑2 − 2𝑒 2 e) No; 𝑏3 is not a perfect square.
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ANSWER
The difference-of-two-square pattern can be used to factor expressions in the form 𝑏2𝑐2 For Example, 4𝑑2 − 81𝑒2 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, 4𝑑2 − 81𝑒2 = 2𝑑 + 9𝑒 2𝑑 − 9𝑒
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FACTORING A DIFFERENCE OF TWO SQUARES
For all number 𝑏 and 𝑐, 𝑏2 − 𝑐2 = (𝑏 + 𝑐)(𝑏 − 𝑐).
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EXAMPLE 4
Factor each expression. a) 𝑦2 − 4 b) 36𝑏2 − 49𝑐2 c) 16𝑦2 − 25 d) 𝑛4 − 𝑜4
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ANSWER
a) 𝑦 + 2 𝑦 − 2 b) (6𝑏 + 7𝑐)(6𝑏 − 7𝑐) c) (4𝑦 + 5)(4𝑦 − 5) d) 𝑛2 + 𝑜2 𝑛2 − 𝑜2 = (𝑛2 + 𝑜2)(𝑛 + 𝑜)(𝑛 − 𝑜)
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EXAMPLE 5
Find each product by using the difference
- f two squares.
a) 31 ∙ 29 b) 17 ∙ 13 c) 34 ∙ 26
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