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Chapter 3: Problem 41

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. $$ x^{3} \geq 9 x^{2} $$

Short Answer

Expert verified
The solution set of the given polynomial inequality is \([0, 9]\).

Step by step solution

01

Rearrange the inequality to zero

Rewrite the inequality by subtracting \(9x^{2}\) from both sides. This gives the inequality \(x^{3} - 9x^{2} \geq 0\).
02

Factor out the polynomial

Factor the left-hand side to simplify the inequality. By factoring out, the inequality becomes \(x^{2}(x - 9) \geq 0\).
03

Find the critical points

Set the factors equal to zero to determine the 鈥榗ritical points鈥� or potential boundaries of the intervals for the solution. This gives \(x_{1} = 0\) and \(x_{2} = 9\) as critical points.
04

Test the intervals

Test the intervals made by the critical points. Choose a test point in each interval \((-鈭�, 0), (0, 9), (9, 鈭�)\) and determine whether the inequality is satisfied. The test points could be -1, 1 and 10, respectively.
05

Identify the solution set

Based on the results in Step 4, identify which intervals satisfy the inequality. Keep in mind that as it is a 鈥榞reater or equal to鈥� inequality, critical points are also included in the solution set.

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Most popular questions from this chapter

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3} $$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=x^{3}-6 x+1, g(x)=x^{3}\)

What are the zeros of a polynomial function and how are they found?

In Exercises 94鈥�97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20$$

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.
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