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Chapter 2: Problem 77

Determine whether the statement is true or false. Justify your answer. The graph of a polynomial function can have infinitely many vertical asymptotes.

Short Answer

Expert verified
The statement is false. A polynomial function can't have any vertical asymptotes because this function does not include any division by a variable.

Step by step solution

01

Recall the definition of Polynomial Function

A Polynomial function is a function that can be expressed in the form of a polynomial. It doesn't include division by a variable.
02

Understand the concept of Vertical Asymptotes

Vertical asymptote exists when we have a function that has points on the graph that the x value (independent variable) is getting infinitesimally close to but never actually reaches, usually due to division by zero in a function.
03

Analyse the possibility of vertical asymptotes in polynomial function

Since Polynomial functions do not include division by a variable, the denominator will never equal to zero. Hence, it is not possible for a polynomial function to have a vertical asymptote. Vertical asymptotes only exist in rational functions when the denominator can equal zero while the numerator does not simultaneously equal zero.

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Most popular questions from this chapter

Find the rational zeros of the polynomial function. $$P(x)=x^{4}-\frac{25}{4} x^{2}+9=\frac{1}{4}\left(4 x^{4}-25 x^{2}+36\right)$$

Use the given zero to find all the zeros of the function. Function \(f(x)=2 x^{3}+3 x^{2}+18 x+27\) Zero \(3 i\)

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{3}{2} x^{2}-\frac{23}{2} x+6=\frac{1}{2}\left(2 x^{3}-3 x^{2}-23 x+12\right)$$

Sketch the graph of each polynomial function. Then count the number of real zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? (a) \(f(x)=-x^{3}+9 x\) (b) \(f(x)=x^{4}-10 x^{2}+9\) (c) \(f(x)=x^{5}-16 x\)
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