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Chapter 3: Problem 82

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

Short Answer

Expert verified
The zeros of a polynomial are the x-values that make the function equal to zero, often representing where the function intersects the x-axis. They are found by setting the polynomial equal to zero and solving for the variable x. With the polynomial function \(f(x) = x^2 - 5x + 6\), solving for x yields the zeros x = 2 and x = 3.

Step by step solution

01

Definition of Polynomial Zeros

Zeros of a polynomial function are the x-values for which the function equals zero. In other words, they are the values of x that satisfy the function. They also correspond to the points where the graph of the function intersects the x-axis.
02

Finding Polynomial Zeros

In order to find the zeros of a polynomial function, you first need to set the polynomial equation equal to zero, and then solve for x. Any numbers that satisfy this equation are the zeros of the function. This process might involve factoring, applying the quadratic formula, or utilizing synthetic division, among other methods.
03

Illustrative Example

Consider the function \(f(x) = x^2 - 5x + 6\). To find the zeros of this polynomial, set the equation equal to zero, \(x^2 - 5x + 6 = 0\), and solve for x. By factoring this as \((x - 2)(x - 3) = 0\), it can be inferred that the zeros for the function are x = 2 and x = 3, as these are the values that satisfy the equation.

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