Find zeros of a polynomial function
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 , use synthetic division to find its zeros.
- Use the Rational Zero Theorem to list all possible rational zeros of the function.
- 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.
- Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.
- 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 .
Solution
The Rational Zero Theorem tells us that if is a zero of , then p is a factor of –1 and q is a factor of 4.
The factors of –1 are and the factors of 4 are , and . The possible values for are , and . 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.
![Synthetic division with 1 as the divisor and {4, 0, -3, -1} as the quotient. Solution is {4, 4, 1, 0}](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201551/Screen-Shot-2015-09-11-at-3.05.49-PM.png)
Dividing by 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. 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 with multiplicity 2.
Analysis of the Solution
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.