Question

How do you find all the rational zeros of a polynomial function?

Answer 1

Find all the rational zeros of a polynomial function:

Rational Root Theorem can be used to find all the rational zeros of the polynomial function.

According to this theorem:

Let the given polynomial be $P\left(x\right)=a_0{x}^n+a_1{x}^{n-1}+...+a_n$ with $a_0,...,a_n$as integers, all rational roots of the form $\frac{p}{q}$ written in the lowest terms will satisfy $P\left(\frac{p}{q}\right)=0$.

Here, $p$ is the divisor of the constant term and $q$ is the divisor of the coefficient of the highest order term of the polynomial function.

Now, substitute each of the possible combinations of $p$ and $q$ as $\frac{p}{q}$ into the polynomial function to get $P\left(\frac{p}{q}\right)=0$.

The values of $\frac{p}{q}$ which gives the value of polynomial function as $0$ are the zeros of the polynomial function.

After getting one zero, the corresponding factor can be divided by the polynomial function to simplify and get more zeros.

Hence, rational zeros of the polynomial function can be found by the Rational Root Theorem.