Solving $x^3-x^2+2=0$

Question

Solve $$x^3-x^2+2=0$$

I feel a bit stupid asking how to solve this, but I'm not sure how to solve this polynomial for $x$. Usually I would use the quadratic formula or factorize the polynomial to find linear factors, but in this case, I'm not sure whether factorization would work or not. How would you solve this?

Answer 1

$$\begin{align}x^3-x^2+2&=(x^3+1)-(x^2-1)\\&=(x+1)(x^2-x+1)-(x+1)(x-1)\\&=(x+1)(x^2-2x+2)\end{align}$$

Answer 2

I would note that $-1$ is a root and then I would divide $x^3-x^2+2$ by $x+1$ and then I would compute the roots of the quotient.

Answer 3

$-1$ is a root divide the polynomial by $x+1$

Answer 4

Since $-1$ is a root, we have $x^3-x^2+2=(x+1)(x^2-2x+2)$. Solving the second quadratic factor we get the remaining roots as $1\pm i$

Answer 5

$$\begin{align}x^3-x^2+2&=x^3+x^2-2x^2-2x+2x+2\\&=x^2(x+1)-2x(x+1)+2(x+1)\\&=(x+1)(x^2-2x+2)\end{align}$$

Answer 6

For general cubics, there isn't anything you can reasonably do by hand, assuming your intention is to express the root in terms of radicals.

Special cases — including any exercise of this sort you will encounter — are amenable to ad-hoc techniques. For example:

• Use guess and check to find a root
• Use the rational root theorem to find a root
• Recognize a pattern in the coefficients that allow you to discover a factor.

Since it's cubic, if you succeed in finding a linear factor, you can divide it out to reduce the equation to a quadratic one, and then use your favorite techniques for those.