Mathematics

Mathematics

Problem

Completely factorise the polynomial q(x)=2x3+x2−7x−6q(x)=2x^{3}+x^{2}-7x-6 over the integers. a) (x+2)(x+1)(2x−3)(x+2)(x+1)(2x-3) b) (x−1)(x−2)(2x+3)(x-1)(x-2)(2x+3) c) (x+1)(x−2)(2x+3)(x+1)(x-2)(2x+3) d) (x−3)(x+1)(2x+2)(x-3)(x+1)(2x+2)
Intuitive explanation

The key is to find one rational root first using the Rational Root Theorem—test small divisors of the constant term divided by divisors of the leading coefficient. Once you find a root and factor it out, you'll reduce this cubic to a quadratic, which is much easier to handle. From there, check whether the remaining quadratic factors nicely over the integers or needs the quadratic formula.

Intuitive explanation

The key is to find one rational root first using the Rational Root Theorem—test small divisors of the constant term divided by divisors of the leading coefficient. Once you find a root and factor it out, you'll reduce this cubic to a quadratic, which is much easier to handle. From there, check whether the remaining quadratic factors nicely over the integers or needs the quadratic formula.

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