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Find the quadratic polynomial that completes the factorization. \newlineu3+1=(u+1)(_____)u^3 + 1 = (u + 1)(\_\_\_\_\_)

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Factor sums and differences of cubesright chevron icon

Full solution

Q. Find the quadratic polynomial that completes the factorization. \newlineu3+1=(u+1)(_____)u^3 + 1 = (u + 1)(\_\_\_\_\_)
  1. Factorization of Sum of Cubes: We know that u3+1u^3 + 1 is a sum of cubes, which factors as (u+1)(u2−u+1)(u + 1)(u^2 - u + 1).
  2. Verification of Factorization: Let's check if our factorization is correct by multiplying (u+1)(u2−u+1)(u + 1)(u^2 - u + 1).(u+1)(u2−u+1)=u3−u2+u+u2−u+1=u3+1(u + 1)(u^2 - u + 1) = u^3 - u^2 + u + u^2 - u + 1 = u^3 + 1.
  3. Completion of Factorization: Now we can see that the quadratic polynomial that completes the factorization is u2−u+1u^2 - u + 1.

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