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Chapter 3: Problem 51

Describe how to use Descartes's Rule of Signs to determine the possible number of negative roots of a polynomial equation.

Short Answer

Expert verified
Descartes's Rule of Signs is applied to negative roots by first substituting the variable in the polynomial equation with its negative equivalent. The possible number of negative real roots is then either the number of times the sign changes in the adjusted polynomial or less by an even number.

Step by step solution

01

Understanding Descartes's Rule of Signs

Descartes's Rule of Signs is a technique used in determining the possible number of positive and negative real roots of a polynomial equation. To apply this rule, we check the sign changes in the polynomial when written in standard form (highest power to lowest). For positives roots, Descartes's Rule states that the number of positive roots of a polynomial equation is equal to the number of sign changes in the equation, or less by an even number. For this exercise, we are interested in negative roots, which requires a slight adjustment.
02

Applying Descartes's Rule for Negative Roots

To figure out the possible number of negative roots, we replace the variable, say x, with (-x) in the polynomial equation and then proceed like we did for positive roots. Again, the number of negative roots is equal to the number of sign changes, or less by an even number.
03

Illustrating with an example

Let's consider an arbitrary polynomial \(P(x) = -2x^4 + 3x^2 - x + 4\). For negative roots, substitute \(x\) with \(-x\) to get \(P(-x) = -2(-x)^4 + 3(-x)^2 - (-x) + 4 = -2x^4 + 3x^2 + x + 4\). This equation has one sign change, so there might be one or no negative root. This means applying Descartes's Rule of Signs enables us to determine the possible number of negative roots of a given polynomial.

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Most popular questions from this chapter

The rational function $$f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\) a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

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. $$f(x)=3 x^{2}-x^{3}$$

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if any, of the functions graph.

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. $$f(x)=-2 x^{4}+4 x^{3}$$

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.
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