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Find the quadratic polynomial that completes the factorization. \newlinex3−64=(x−4)(_____)x^3 - 64 = (x - 4)(\_\_\_\_\_)

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Full solution

Q. Find the quadratic polynomial that completes the factorization. \newlinex3−64=(x−4)(_____)x^3 - 64 = (x - 4)(\_\_\_\_\_)
  1. Identify Difference of Cubes: We know that x3−64x^3 - 64 is a difference of cubes, which factors as (x−a)(x2+ax+a2)(x - a)(x^2 + ax + a^2), where aa is the cube root of 6464.
  2. Substitute Cube Root: The cube root of 6464 is 44, so we substitute a=4a = 4 into the formula: (x−4)(x2+4x+16)(x - 4)(x^2 + 4x + 16).
  3. Expand Factored Expression: Now we check if (x−4)(x2+4x+16)(x - 4)(x^2 + 4x + 16) expands to x3−64x^3 - 64.
  4. Simplify Expanded Expression: Expanding (x−4)(x2+4x+16)(x - 4)(x^2 + 4x + 16) gives us x3+4x2+16x−4x2−16x−64x^3 + 4x^2 + 16x - 4x^2 - 16x - 64.
  5. Simplify Expanded Expression: Expanding (x−4)(x2+4x+16)(x - 4)(x^2 + 4x + 16) gives us x3+4x2+16x−4x2−16x−64x^3 + 4x^2 + 16x - 4x^2 - 16x - 64. Simplify the expression: x3+4x2−4x2+16x−16x−64=x3−64x^3 + 4x^2 - 4x^2 + 16x - 16x - 64 = x^3 - 64.

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