Factoring Sums or Differences of Cubes

Write an equivalent expression by factoring: x3  27 Example Write an equivalent expression by factoring: x3  27 Solution We observe that x3  27 = x3  (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)(x2 + 3x + 9) Thus, x3  27 = (x  3)(x2 + 3x + 9) Square the first term: x2. Multiply the terms and then change the sign: 3x. Square the second term: (3)2, or 9. Same Opposite Always Positive = SOAP

Write an equivalent expression by factoring: x3 + 27 Example Write an equivalent expression by factoring: x3 + 27 Solution We observe that x3 + 27 = x3 + (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)(x2  3x + 9) Thus, x3 + 27 = (x + 3)(x2  3x + 9) Square the first term: x2. Multiply the terms and then change the sign: 3x. Square the second term: (3)2, or 9. Same Opposite Always Positive = SOAP

Example Factor: x3 + 27 Solution We have x3 + 27 = x3 + (3)3 = (x + 3) (x2  x  3 + 32) A3 + B3 = (A + B) (A2  A B + B2) The factorization is: (x + 3)(x2  3x + 9) Example Factor: 27x3  8y3 Solution We have 27x3  8y3 = (3x)3  (2y)3 = (3x  2y)[(3x)2 + (3x)(2y) + (2y)2] The factorization is: (3x  2y)(9x2 + 6xy + 4y2)

Example Factor: 3ab4 + 192a4b10 Solution First we look for a greatest common factor: 3ab4 + 192a4b10 = 3ab4(1 + 64a3b6) = 3ab4[13 + (4ab2)3] = 3ab4(1 + 4ab2)(1  4ab2 + 16a2b4) The factorization is: 3ab4(1 + 4ab2)(1  4ab2 + 16a2b4)

Solving Polynomial Equations Example: Solve x3 = 64. Solution – Algebraic Rewrite to get 0 on one side and factor: x3 = 64 x3 – 64 = 0 Subtracting 64 from both sides, set equal to zero (x – 4)(x2 + 4x + 16) = 0 Factoring x2 + 4x + 16 does not factor using real coefficients. x – 4 = 0 or x2 + 4x + 16 = 0 x = 4 We have one solution, x = 4. Graphical Solution The solution is x = 4.

Solution – Graphical Solution Example: Solution – Graphical Solution The solutions are: x = 0.070, x = 2.213, or x = 3.217.