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Find the quadratic polynomial that completes the factorization. \newliner3−512=(r−8)(_____)r^3 - 512 = (r - 8)(\_\_\_\_\_)

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Factor sums and differences of cubesright chevron icon

Full solution

Q. Find the quadratic polynomial that completes the factorization. \newliner3−512=(r−8)(_____)r^3 - 512 = (r - 8)(\_\_\_\_\_)
  1. Recognize as difference of cubes: To factor r3−512r^3 - 512, recognize it as a difference of cubes: r3−83r^3 - 8^3.
  2. Apply formula for factoring: The formula for factoring a difference of cubes is a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2).
  3. Factor using aa and bb: Apply the formula with a=ra = r and b=8b = 8: (r−8)(r2+8r+64)(r - 8)(r^2 + 8r + 64).
  4. Check the multiplication: Check the multiplication: r - \(8)(r^22 + 88r + 6464) = r^33 + 88r^22 + 6464r - 88r^22 - 6464r - 512512\
  5. Simplify the expression: Simplify the expression: r3+8r2+64r−8r2−64r−512=r3−512r^3 + 8r^2 + 64r - 8r^2 - 64r - 512 = r^3 - 512.

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