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Chapter 3: Problem 78

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

Short Answer

Expert verified
The rational zeros of the given polynomial function are -2 and 3.

Step by step solution

01

Rewrite the Polynomial

First, rewrite the polynomial in its simplified form: \(f(x) = \frac{1}{2}(2x^3 - 3x^2 - 23x + 12)\) which equals \(f(x) = x^3 - \frac{3}{2}x^2 - \frac{23}{2}x + 6\)
02

Apply the Rational Root Theorem

Apply the Rational Root Theorem. Identify the leading term and the constant term of the polynomial. For the polynomial \(x^3 - \frac{3}{2}x^2 - \frac{23}{2}x + 6\), the leading coefficient is 1 and the constant term is 6. According to the theorem, any rational roots of the polynomial would be factors of 6 divided by factors of 1. The factors of 6 are 卤1, 卤2, 卤3, 卤6, hence the possible rational roots of the polynomial are also 卤1, 卤2, 卤3, 卤6.
03

Test the Possible Rational Roots

Substitute each possible rational root into the polynomial and check if the result is zero. When trying 卤1, 卤2, 卤3, and 卤6, it can be found that -2 and 3 are roots of the polynomial since \(f(-2)=0\) and \(f(3)=0\).

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Most popular questions from this chapter

A graph of \(y=f(x)\) is shown, where \(f(x)=2 x^{5}-3 x^{4}+x^{3}-8 x^{2}+5 x+3\) and \(f(-x)=-2 x^{5}-3 x^{4}-x^{3}-8 x^{2}-5 x+3\). (a) How many negative real zeros does \(f\) have? Explain. (b) How many positive real zeros are possible for \(f ?\) Explain. What does this tell you about the eventual right-hand behavior of the graph? (c) Is \(x=-\frac{1}{3}\) a possible rational zero of \(f ?\) Explain. (d) Explain how to check whether \(\left(x-\frac{3}{2}\right)\) is a factor of \(f\) and whether \(x=\frac{3}{2}\) is an upper bound for the real zeros of \(f\).

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=x^{4}-4 x^{3}+15\) Upper bound: \(x=4\) Lower bound: \(x=-1\)

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.

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((g-f)(3)\)

Find all real zeros of the polynomial function. $$h(x)=x^{5}+5 x^{4}-5 x^{3}-15 x^{2}-6 x$$
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