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Find the quadratic polynomial that completes the factorization.\newlinec3−216=(c−6)(_____)c^3 - 216 = (c - 6)(\_\_\_\_\_)

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Full solution

Q. Find the quadratic polynomial that completes the factorization.\newlinec3−216=(c−6)(_____)c^3 - 216 = (c - 6)(\_\_\_\_\_)
  1. Find Cube Root of 216216: We know that c3−216c^3 - 216 is a difference of cubes, which factors as (c−a)(c2+ac+a2)(c - a)(c^2 + ac + a^2) where a3=216a^3 = 216. So first, find the cube root of 216216.\newlineCalculation: a=216(1/3)=6a = 216^{(1/3)} = 6
  2. Plug in Value for Difference of Cubes: Now plug a=6a = 6 into the formula for the difference of cubes.\newlineCalculation: (c−6)(c2+6c+36)(c - 6)(c^2 + 6c + 36)
  3. Check Expanded Form: Check if the expanded form of (c−6)(c2+6c+36)(c - 6)(c^2 + 6c + 36) equals c3−216c^3 - 216.\newlineCalculation: (c−6)(c2+6c+36)=c3+6c2+36c−6c2−36c−216=c3−216(c - 6)(c^2 + 6c + 36) = c^3 + 6c^2 + 36c - 6c^2 - 36c - 216 = c^3 - 216

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