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Find the quadratic polynomial that completes the factorization. \newlinex3+512=(x+8)(_____)x^3 + 512 = (x + 8)(\_\_\_\_\_)

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Factor sums and differences of cubesright chevron icon

Full solution

Q. Find the quadratic polynomial that completes the factorization. \newlinex3+512=(x+8)(_____)x^3 + 512 = (x + 8)(\_\_\_\_\_)
  1. Identify Sum of Cubes: We know that x3+512x^3 + 512 is a sum of cubes because 512512 is 838^3. The sum of cubes formula is a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2). Here, a=xa = x and b=8b = 8.
  2. Apply Sum of Cubes Formula: Apply the sum of cubes formula: x3+83=(x+8)(x2−x∗8+82)x^3 + 8^3 = (x + 8)(x^2 - x\ast8 + 8^2).
  3. Calculate Terms: Calculate the terms inside the parentheses: x2−x×8+82=x2−8x+64x^2 - x\times 8 + 8^2 = x^2 - 8x + 64.
  4. Find Quadratic Polynomial: So, the quadratic polynomial that completes the factorization is x2−8x+64x^2 - 8x + 64.

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