If $x^2 + ax - b$ is a factor of $x^3 - 2bx^2 + ax -6$, show that $a = -2b -6/b$.

Question

If $x^2 + ax - b$ is a factor of $x^3 - 2bx^2 + ax -6$, show that $a = -2b -6/b$.

Answer 1

Suppose $x^3-2bx^2+ax-6$ has the factor $x^2+ax-b$. Suppose the other factor is $x-q$. Then:

$$(x-q)(x^2+ax+b)=x^3-2bx^2+ax-6$$

But by expanding:

$$(x-q)(x^2+ax+b)=x^3-qx^2+ax^2-qax+bx-qb=x^3+(a-q)x^2+(b-aq)x-bq$$

By comparing coefficients we have the equations:

$$a-q=-2b, \quad b-aq = a, \quad -bq=6$$

Hence $q=a+2b$ and $6 = -b(a+2b)$. This will reduce to $a = -2b-6/b$.