Finding the Zeros of a Polynomial Function with Repeated Real Zeros

The Rational Zero Theorem tells us that if $\frac{p}{q}$ is a zero of $f(x),$ then $p$ is a factor of –1 and $q$ is a factor of 4.

The factors of $\mathrm{–1}$ are $\mathrm{\pm 1}$ and the factors of $4$ are $\mathrm{\pm 1},\mathrm{\pm 2},$ and $\mathrm{\pm 4.}$ The possible values for $\frac{p}{q}$ are $\mathrm{\pm 1},\pm \frac{1}{2},$ and $\pm \frac{1}{4}.$ These are the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let’s begin with 1.

Dividing by $(x-1)$ gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as

The quadratic is a perfect square. $f(x)$ can be written as

We already know that 1 is a zero. The other zero will have a multiplicity of 2 because the factor is squared. To find the other zero, we can set the factor equal to 0.

The zeros of the function are 1 and $-\frac{1}{2}$ with multiplicity 2.