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Chapter 2: Problem 5

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$f(x)=4 x^{4}-x^{3}+5 x^{2}-2 x-6$$

Short Answer

Expert verified
The possible rational zeros of the function \( f(x) = 4x^{4} - x^{3} + 5x^{2} - 2x - 6 \) are -6, -3, -2, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 3, and 6.

Step by step solution

01

Identifying the Constant and Leading Coefficients

The first step is to identify the constant term and the leading coefficient of the polynomial function. Here, the constant term \( a_0 \) is -6 and the leading coefficient \( a_n \) is 4.
02

Listing Factors

The next step is to list all the factors of \( a_0 \) and \( a_n \). The factors of -6 are -6, -3, -2, -1, 1, 2, 3, and 6, while the factors of 4 are -4, -2, -1, 1, 2, and 4.
03

Formulating the Rational Zeros

The Rational Zero Theorem allows us to find all potential rational zeros, which are of the form \( p/q \). These are obtained by dividing every factor of \( a_0 \) by every factor of \( a_n \). This results in the list of possible zeros: -6, -3, -2, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 3, and 6.

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Most popular questions from this chapter

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}-1}{x^{2}-9}$$

What is a rational inequality?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a polynomial inequality that has a value for which the polynomial function is undefined.

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+2$$

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.
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