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Find the quadratic polynomial that completes the factorization. \newlineq3+216=(q+6)(_____)q^3 + 216 = (q + 6)(\_\_\_\_\_)

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Full solution

Q. Find the quadratic polynomial that completes the factorization. \newlineq3+216=(q+6)(_____)q^3 + 216 = (q + 6)(\_\_\_\_\_)
  1. Recognize formula: Recognize the sum of cubes formula: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2). Here, a=qa = q and b=6b = 6.
  2. Apply formula: Apply the formula: q3+216=q3+63=(q+6)(q2−q⋅6+62)q^3 + 216 = q^3 + 6^3 = (q + 6)(q^2 - q\cdot6 + 6^2).
  3. Calculate terms: Calculate the terms: q2−q×6+62=q2−6q+36q^2 - q\times 6 + 6^2 = q^2 - 6q + 36.
  4. Write factorization: Write the complete factorization: q3+216=(q+6)(q2−6q+36)q^3 + 216 = (q + 6)(q^2 - 6q + 36).

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