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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 function are the x-values that make the polynomial equal to zero. To find the zeros, we usually set the polynomial equal to zero and solve for x, which often requires factoring. In the example given, the zeros of the polynomial \(f(x) = x^2 - 5x + 6\) are \(x = 2\) and \(x = 3\).

Step by step solution

01

Definition of Zeros of a Polynomial Function

The zeros of a polynomial function are the x-values that make the polynomial equal to zero. These values are also called roots. If we have a polynomial \(f(x)\), then \(x = a\) is a zero of the polynomial if \(f(a) = 0\).
02

Factoring the Polynomial

To find the zeros, set the polynomial equal to zero and solve for x. This usually involves factoring. For example, if our polynomial is \(f(x) = x^2 - 5x + 6\), we set it equal to zero and factor: \(0 = x^2 - 5x + 6\) which after factoring becomes \(0 = (x - 2)(x - 3)\).
03

Solving for X

Take each factor that includes a variable and set it equal to zero, then solve. Therefore, when \(x - 2 = 0\), \(x = 2\) and when \(x - 3 = 0\), \(x = 3\). Those are the zeros of the polynomial.

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-3(x-1)^{2}\left(x^{2}-4\right)$$

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$

Describe a strategy for graphing a polynomial function. In your description, mention intercepts, the polynomial鈥檚 degree, and turning points.
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