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Find the binomial that completes the factorization. \newlinew3+512=(‾)(w2−8w+64)w^3 + 512 = (\underline{\hspace{3cm}})(w^2 - 8w + 64)

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Full solution

Q. Find the binomial that completes the factorization. \newlinew3+512=(‾)(w2−8w+64)w^3 + 512 = (\underline{\hspace{3cm}})(w^2 - 8w + 64)
  1. Recognize Perfect Cube: Recognize that 512512 is a perfect cube, 512=83512 = 8^3.
  2. Use Sum of Cubes Formula: Use the sum of cubes formula: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).
  3. Identify aa and bb: Identify a=wa = w and b=8b = 8, since w3+83w^3 + 8^3 is the expression we have.
  4. Plug into Formula: Plug aa and bb into the sum of cubes formula: w3+83=(w+8)(w2−w⋅8+82)w^3 + 8^3 = (w + 8)(w^2 - w\cdot8 + 8^2).
  5. Simplify Binomial and Trinomial: Simplify the binomial and trinomial: (w+8)(w2−8w+64)(w + 8)(w^2 - 8w + 64).

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