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Find the binomial that completes the factorization. \newlineq3−r3=(‾)(q2+qr+r2)q^3 - r^3 = (\underline{\hspace{3em}})(q^2 + qr + r^2)

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Full solution

Q. Find the binomial that completes the factorization. \newlineq3−r3=(‾)(q2+qr+r2)q^3 - r^3 = (\underline{\hspace{3em}})(q^2 + qr + r^2)
  1. Factorize given expression: The given expression is a difference of cubes, which factors as (a3−b3)=(a−b)(a2+ab+b2)(a^3 - b^3) = (a - b)(a^2 + ab + b^2).
  2. Identify values of aa and bb: Identify aa and bb in the expression q3−r3q^3 - r^3; here, a=qa = q and b=rb = r.
  3. Write binomial factor: Write the binomial factor (a−b)(a - b) using the identified values of aa and bb; (q−r)(q - r).
  4. Check factorization by multiplication: Check the factorization by multiplying (q−r)(q - r) with (q2+qr+r2)(q^2 + qr + r^2) to see if it equals q3−r3q^3 - r^3.\newline(q−r)(q2+qr+r2)=q3−r3(q - r)(q^2 + qr + r^2) = q^3 - r^3.

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