Factor x^3+(2a+1)x^2+(a^2+2a-1)x+(a^2-1)

Question

anonymous · Answer

$$x^3+(2a+1)x^2+(a^2+2a-1)x+(a^2-1)$$

whpalmer4 · Answer

Here's a somewhat roundabout hint:

when you have a polynomial with roots $a\pm bi$ and need to construct the polynomial, you multiply 
$$(x-a-bi)(x-a+bi) = x^2-2ax +a^2+b^2$$the sign of the $b^2$ term changes because of the $i^2=-1$ so let's consider the related multiplication
$$(x+a-b)(x+a+b) = x^2 + ax + bx + ax + a^2 + a b - bx - ab - b^2$$$$\qquad=x^2+2ax + a^2 -b ^2$$
Notice that the constant term here is $a^2 -b^2$, and the constant term in our polynomial to be factored could be written the same way, if $b =1$
This suggests that two of our factors might be $(x+a-1)(x+a+1)$

Now multiplying those two terms together only gives you a polynomial with an $x^2$, not an $x^3$, but hopefully it is clear upon consideration that if we multiply by another binomial, we'll get that $x^3$ term we need, and we could find that binomial by polynomial long division, if necessary.