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Chapter 3: Problem 116

Determine whether the statement is true or false. Justify your answer. It is possible for a polynomial with an even degree to have a range of \((-\infty, \infty)\)

Short Answer

Expert verified
The statement is false. An even degree polynomial cannot have a range of all real numbers.

Step by step solution

01

Understanding Polynomial Properties

A polynomial function of even degree will either open upwards on both ends if the leading coefficient is positive, or downwards on both ends if the leading coefficient is negative. This is because for even-degree functions, as \(x\) tends toward positive or negative infinity, \(f(x)\) also tends toward positive or negative infinity depending on the sign of the leading coefficient.
02

Assessing Range

From the above understanding, we can infer that an even-degree polynomial will not cover the entire set of real numbers, i.e., it will not have a range of \((-\infty, \infty)\). If the leading coefficient is positive, the range of function will be \([k, \infty)\) where \(k\) is the minimum value of function. If leading coefficient is negative, the range will be \((-\infty, k]\) where \(k\) is the maximum value of the function.
03

Conclusion

From these assessments, the statement that a polynomial with an even degree can have a range of \((-\infty, \infty)\) is FALSE. An even-degree polynomial function will either have a minimum value and extend to positive infinity or have a maximum value and extend to negative infinity, but it cannot extend in both directions on the y-axis.

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Most popular questions from this chapter

The table shows the numbers \(S\) of cellular phone subscriptions per 100 people in the United States from 1995 through 2012 . The data can be approximated by the model \(S=-0.0223 t^{3}+0.825 t^{2}-3.58 t+12.6\) \(5 \leq t \leq 22\) where \(t\) represents the year, with \(t=5\) corresponding to 1995 (a) Use a graphing utility to plot the data and graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the Remainder Theorem to evaluate the model for the year \(2020 .\) Is the value reasonable? Explain.

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Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-1$$
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