How can you determine if a root of a polynomial is a repeated one?

Question

Let's say we have the polynomial $x^3 − 12x − 16$, and we know there are 2 roots, $x = -2$, and $x = 4$, and one of those roots appears twice.

How can we determine which of those two roots is the repeated one?

Answer 1

With algebra, you can factor (or do something equivalent, like synthetic division or polynomial long division). Like here you can compute that $\frac{x^3-12x-16}{(x+2)(x-4)}=x+2$ for $x \neq -2,4$.

Another option with algebra is guess-and-check: you can multiply out $(x+2)^2(x-4)$ and $(x-4)^2(x+2)$ and see which one matches.

With calculus, you can take the derivative and plug in the roots; if you get zero at a root, then the point is at least a double root. Higher roots can be checked by taking more derivatives (e.g. $p(0)=p'(0)=p''(0)=0$ with $p'''(0) \neq 0$ means a triple root).

Answer 2

You differentiate your polynomial, which will give you $3x^2-12$. Since $-2$ is a root of it, it's $-2$ which is the double root.

Answer 3

You could use Vieta's formulas.

For this polynomial, the sum of the roots must be $0,$ since that is the coefficient of $x^2$;

$-2+-2+4=0$ and $4+-2+4\ne0,$ so the answer is that $-2$ is repeated.