Problem 61 - College Algebra

Question

What are the zeros of a polynomial function and how are they found?

Step-by-Step Solution

Answer

Zeros of a polynomial function are the x-values where the function equals zero. They are found by using various methods including isolation of terms, quadratic formula, synthetic division, or factoring, based on the degree of the polynomial.

Step 1: Understanding Zeros of Polynomial

The zeros of a polynomial are those values of the variable which make the polynomial equal to zero. In other words, if inserting a certain x-value into the equation makes the outcome zero, then that x-value is a zero of the polynomial.

Step 2: Methods to Find Zeros

There are different methods to find the zeros depending on the degree of the polynomial. For linear polynomials (degree 1), just simplifying the equation to the term containing the variable isolated on one side will give the solution. For quadratic polynomials (degree 2), quadratic formula can be used. For higher degree polynomials, one might need to use synthetic division or factoring.

Step 3: Applying Methods

Apply these methods depending on the degree of your polynomial function. If it is a simple linear equation, isolate the variable term to find the zero. If it is a quadratic equation, use the quadratic formula \(-\frac{b}{2a} \pm \sqrt{(\frac{b}{2a})^2 - a}\), where a, b, and c are coefficients in a quadratic equation in the form \(ax^2 + bx + c = 0\). For higher degree equations, use synthetic division or factoring, methods which generally require a bit more of computational work.