/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Solve each polynomial inequality... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 40

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}-x^{2}+9 x-9>0 $$

Short Answer

Expert verified
The solution set in interval notation is \((1.0047, +\infty)\)

Step by step solution

01

Factorize the cubic equation

The first step is to factorize the equation \(x^{3}-x^{2}+9x -9 = 0\). However, this specific polynomial doesn't have a simple factorization. Hence, we can use the Rational Root Theorem or approximations to find its roots.
02

Find the roots

The root of this cubic equation can be found by graphical analysis or using a calculator. In this case, the roots of the equation \(x^{3}-x^{2}+9x -9 = 0\) are approximately \(x \approx 1.0047\), \(x \approx -3.3971-0.7104i\), and \(x \approx -3.3971+0.7104i\). Considering we're graphing on a real number line, we only need the real root, which is \(x \approx 1.0047\).
03

Determine the intervals

We can now establish intervals for the inequality. We have one real root, dividing the line into two intervals. These are \((-\infty, 1.0047)\) and \((1.0047, +\infty)\). Choose a test point in each interval to check if the inequality is satisfied.
04

Check the intervals

We need to substitute the test points into the inequality. For the first interval, let's pick \(x=0\). Substituting that into \(x^{3}-x^{2}+9x -9 > 0\), we find that it's less than zero, therefore this interval isn't part of the solution set. For the second interval, let's pick \(x=2\), doing the same we find that it's greater than zero, therefore this interval is part of the solution set.
05

Express in interval notation

Finally, express the solution as an interval, taking into account the inequality is 'greater than' and not 'greater than or equal to'. This indicates that we should use a parenthesis, not square brackets. Our solution in interval notation becomes \((1.0047, +\infty)\).

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. 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 maximum number of turning points to check whether it is drawn correctly. $$f(x)=3 x^{2}-x^{3}$$

In Exercises 104鈥�107, 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 function with origin symmetry can rise to the left and rise to the right.

Use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. \(f(x)=-x^{4}+8 x^{3}+4 x^{2}+2\)

If f is a polynomial function, and f(a) and f(b) have opposite signs, what must occur between a and b? If f(a) and f(b) have the same sign, does it necessarily mean that this will not occur? Explain your answer.

Rewrite \(4-5 x-x^{2}+6 x^{3}\) in descending powers of \(x\).
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