2122.11 – Factoring a Cubic

If a≠ba \neq b, a3−b3=19x3a^3 - b^3 = 19x^3, and a−b=xa - b = x, which of the following conclusions is correct?

  1. a=3xa = 3x

  2. a=3xa = 3x or a=−2xa = -2x

  3. a=−3xa = -3x or a=2xa = 2x

  4. a=3xa = 3x or a=2xa = 2x

  5. a=2xa = 2x

Solution

Given a≠ba \neq b, a3−b3=19x3a^3 - b^3 = 19x^3, and a−b=xa - b = x, we need to find aa in terms of xx because that is the form of all the given answers. Therefore, bb must go, and so we note that b=a−xb = a - x. We next take a deep breath.

a3−b3=(a−b)(a2+ab+b2)19x3=x(a2+a(a−x)+(a−x)2)=x(a2+a2−ax+a2−2ax+x2)=x(3a2−3ax+x2)=3a2x−3ax2+x30=3a2x−3ax2−18x3=3x(a2−ax−6x2)=3x(a−3x)(a+2x)\begin{aligned}a^3 - b^3 &= (a - b)(a^2 + ab + b^2) \\19x^3 &= x (a^2 + a (a - x) + (a - x)^2) \\&= x (a^2 + a^2 - ax + a^2 - 2ax + x^2) \\&= x (3a^2 - 3a^x + x^2) \\&= 3a^2x - 3ax^2 + x^3 \\0 &= 3a^2x - 3ax^2 - 18x^3 \\&= 3x (a^2 - ax - 6x^2) \\&= 3x (a - 3x)(a + 2x)\end{aligned}

So a =3xa = 3x or a =−2xa = -2x. This is answer (b). Whew.