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Find the quadratic polynomial that completes the factorization. \newlinex3−y3=(x−y)(_____)x^3 - y^3 = (x - y)(\_\_\_\_\_)

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Factor sums and differences of cubesright chevron icon

Full solution

Q. Find the quadratic polynomial that completes the factorization. \newlinex3−y3=(x−y)(_____)x^3 - y^3 = (x - y)(\_\_\_\_\_)
  1. Identify Difference of Cubes: To factor x3−y3x^3 - y^3, we know it's a difference of cubes. The formula for factoring a3−b3a^3 - b^3 is (a−b)(a2+ab+b2)(a - b)(a^2 + ab + b^2).
  2. Apply Factorization Formula: Let's apply the formula to x3−y3x^3 - y^3. Here, a=xa = x and b=yb = y. So, the factorization is (x−y)(x2+xy+y2)(x - y)(x^2 + xy + y^2).
  3. Verify Factorization: Now we check if the factorization is correct by expanding (x−y)(x2+xy+y2)(x - y)(x^2 + xy + y^2) to see if we get x3−y3x^3 - y^3 back.(x−y)(x2+xy+y2)=x3+x2y+xy2−yx2−xy2−y3(x - y)(x^2 + xy + y^2) = x^3 + x^2y + xy^2 - yx^2 - xy^2 - y^3=x3−y3.= x^3 - y^3.

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