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Find the binomial that completes the factorization. \newlinev3−512=(‾)(v2+8v+64)v^3 - 512 = (\underline{\hspace{3cm}})(v^2 + 8v + 64)

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Full solution

Q. Find the binomial that completes the factorization. \newlinev3−512=(‾)(v2+8v+64)v^3 - 512 = (\underline{\hspace{3cm}})(v^2 + 8v + 64)
  1. Recognize Perfect Cube: Recognize that 512512 is a perfect cube, 512=83512 = 8^3. So, we can write v3−512v^3 - 512 as v3−83v^3 - 8^3.
  2. Use Difference of Cubes Formula: Use the difference of cubes formula: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here, a=va = v and b=8b = 8.
  3. Apply Formula: Apply the formula to v3−83v^3 - 8^3 to get (v−8)(v2+8v+64)(v - 8)(v^2 + 8v + 64).

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