Is there an easy way to factor $$f(x)=-x^3+x^2-2\;?$$
I have checked step-by-step calculators that all use theorems I am not very familiar with.
It doesn't seem like a sum or difference of cubes, and grouping is not an option. I tried factoring out the $-x^2$ $$f(x)=-x^2(x-1)-2$$ but that didn't get me anywhere. I am trying to find the roots to graph this function. How do I approach this?
Best Answer
$$-x^3+x^2-2=-1-x^3+x^2-1$$ $$=-(1+x^3)+(x^2-1)$$ $$=-(1+x)(1-x+x^2)+(x-1)(x+1)$$ $$=(x+1)(x-1-(1-x+x^2))$$ $$=(x+1)(-x^2+2x-2)$$
Edit: At the fourth step, there is a factor of $x+1$ in both terms. So I factor this out of the expression as follows: $$-(1+x)(1-x+x^2)+(x-1)(x+1)=(x+1)[-(1-x+x^2)]+(x+1)(x-1)$$ $$=(x+1)[-(1-x+x^2)+(x-1)]$$ $$=(x+1)(-1+x-x^2+x-1)$$ $$=(x+1)(-x^2+2x-2)$$