Cubic factoring question

Question

I'm trying to figure out how a colleague factored an expression. I don't get how:

$$a^3+a^2b-(b+1)=(a-1)[a^2+a(b+1)+(b+1)]$$

Multiplying the result I see it's true, but not sure how he got there..is this a general rule for cubic expressions of this form?

Thanks,

P

Answer 1

\begin{align*} a^3+a^2b-(b+1) &=a^3-a^2+a^2+a^2b-(b+1) \\&=a^3-a^2+a^2(b+1)-(b+1) \\&=(a-1)a^2+(a^2-1)(b+1) \\&=(a-1)a^2+(a-1)(a+1)(b+1) \\&=(a-1)(a^2+(a+1)(b+1)) \\&=(a-1)(a^2+a(b+1)+(b+1)) \end{align*}

Answer 2

I'd do it like this, since it is linear in $b$:$$a^3+a^2b-(b+1)=(a^2-1)b+(a^3-1) = (a-1)\left ((a+1)b+a^2+a+1\right) =$$ if you want the inner bracket expressed as a polynomial in $a$ $$(a-1)\left(a^2+(b+1)a+(b+1)\right)$$

Note that though the common factor here is obvious, in a more complex case $p(a)b+q(a)$ it would be possible to use the Euclidean algorithm to find any common factors between $p(a)$ and $q(a)$.

Alternatively note that $a=1$ is a root of the original expression and use polynomial division to extract the factor $a-1$.