Finding the Zeros of Polynomial Functions

The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function.


HOW TO

Given a polynomial function f, use synthetic division to find its zeros.

1. Use the Rational Zero Theorem to list all possible rational zeros of the function.

2. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.

3. Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.

4. Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.


EXAMPLE 5

Finding the Zeros of a Polynomial Function with Repeated Real Zeros

Find the zeros of f(x)=4 x^{3}-3 x-1.


Solution

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.

\begin{aligned} \frac{p}{q} &=\frac{\text { factor of constant term }}{\text { factor of leading coefficient }} \\ &=\frac{\text { factor of }-1}{\text { factor of } 4} \end{aligned}

The factors of –1 are ±1 and the factors of 4 are \pm 1, \pm 2, and \pm 4. The possible values for \frac{p}{q} are \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

(x-1)\left(4 x^{2}+4 x+1\right)

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

(x-1)(2 x+1)^{2}

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.

\begin{aligned} 2 x+1 &=0 \\ x &=-\frac{1}{2} \end{aligned}

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


Analysis

Look at the graph of the function f in Figure 1. Notice, at x=−0.5, the graph bounces off the x-axis, indicating the even multiplicity (2,4,6…) for the zero −0.5. At x=1, the graph crosses the x-axis, indicating the odd multiplicity (1,3,5…) for the zero x=1.

Figure 1