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Published byRonaldo Bluitt Modified about 1 year ago

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Factoring Sums or Differences of Cubes

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Example Write an equivalent expression by factoring: x 3 27 Solution We observe that x 3 27 = x 3 (3) 3 In one set of parentheses, we write the first cube root, x minus the second cube root, 3: (x 3)( ) To get the other factor, we think of x – 3 and do the following: (x 3)(x 2 + 3x + 9) Thus, x 3 27 = (x 3)(x 2 + 3x + 9) Square the first term: x 2. Multiply the terms and then change the sign: 3x. Square the second term: ( 3) 2, or 9. SameOpposite Always Positive= SOAP

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Example Write an equivalent expression by factoring: x Solution We observe that x = x 3 + (3) 3 In one set of parentheses, we write the first cube root, x plus the second cube root, 3: (x + 3)( ) To get the other factor, we think of x + 3 and do the following: (x + 3)(x 2 3x + 9) Thus, x = (x + 3)(x 2 3x + 9) Square the first term: x 2. Multiply the terms and then change the sign: 3x. Square the second term: (3) 2, or 9. SameOpposite Always Positive= SOAP

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Example Factor: x Solution We have x = x 3 + (3) 3 = (x + 3) (x 2 x ) A 3 + B 3 = (A + B) (A 2 A B + B 2 ) The factorization is: (x + 3)(x 2 3x + 9) Example Factor: 27x 3 8y 3 Solution We have 27x 3 8y 3 = (3x) 3 (2y) 3 = (3x 2y)[(3x) 2 + (3x)(2y) + (2y) 2 ] The factorization is: (3x 2y)(9x 2 + 6xy + 4y 2 )

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Example Factor: 3ab a 4 b 10 Solution First we look for a greatest common factor: 3ab a 4 b 10 = 3ab 4 (1 + 64a 3 b 6 ) = 3ab 4 [1 3 + (4ab 2 ) 3 ] = 3ab 4 (1 + 4ab 2 )(1 4ab a 2 b 4 ) The factorization is: 3ab 4 (1 + 4ab 2 )(1 4ab a 2 b 4 )

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Solving Polynomial Equations Example: Solve x 3 = 64. Solution – Algebraic Rewrite to get 0 on one side and factor: x 3 = 64 x 3 – 64 = 0 Subtracting 64 from both sides, set equal to zero (x – 4)(x 2 + 4x + 16) = 0 Factoring x 2 + 4x + 16 does not factor using real coefficients. x – 4 = 0 or x 2 + 4x + 16 = 0 x = 4 We have one solution, x = 4. Graphical Solution The solution is x = 4.

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Example: Solution – Graphical Solution The solutions are: x = 0.070, x = 2.213, or x =

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