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Chapter 3: Problem 64

Can the graph of a polynomial function have no \(x\) -intercepts? Explain.

Short Answer

Expert verified
Yes, it is possible for the graph of a polynomial function to have no x-intercepts. This can occur if the function represents a constant that is not equal to zero or if the function is shifted vertically in a way that it does not cross the x-axis.

Step by step solution

01

Define an x-intercept

An x-intercept is an x-coordinate where the function's value is zero, represented by points on the graph where it crosses the x-axis.
02

Consider a constant polynomial

Consider a constant polynomial function, such as f(x) = c where c is not equal to zero. This function's graph is a horizontal line that never touches or crosses the x-axis, thus it does not have any x-intercepts.
03

Consider Zero Polynomial Function

Consider another scenario where a polynomial function is f(x) = 0, this function has every value of x as its x-intercept.
04

Discuss other polynomial functions

Other polynomial functions of higher degrees (linear, quadratic, cubic, etc.) will generally have one or more x-intercepts, unless they are shifted vertically in such a way that they do not cross the x-axis. Thus, it is possible for a polynomial function to have no x-intercepts, although this is not common.

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Most popular questions from this chapter

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=3 x^{5}-2 x^{4}+6 x^{3}-4 x^{2}-24 x+16 $$

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) 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 fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=6 x^{3}-9 x-x^{5}$$

Why is a third-degree polynomial function with a negative leading coefficient not appropriate for modeling nonnegative real-world phenomena over a long period of time?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) 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 fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2 x^{4}+4 x^{3}$$

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) 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 fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=3 x^{2}-x^{3}$$
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