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Chapter 2: Problem 34

Solve the inequality. Then graph the solution set. $$x^{4}(x-3) \leq 0$$

Short Answer

Expert verified
The solution to the inequality is \(0 \leq x \leq 3\). The graph will show a solid line between these points, inclusive.

Step by step solution

01

Find the roots of the polynomial

Set the polynomial \(x^{4}(x-3) = 0\). Solve for x to find the roots, which are \(x=0\) (from \(x^{4}=0\)) and \(x=3\) (from \(x-3=0\)).
02

Determine the intervals

The roots divide the x-axis into three intervals: \(x < 0\), \(0 < x < 3\), and \(x > 3\).
03

Perform a test in each interval

Test a number from each interval in the inequality \(x^{4}(x-3) \leq 0\) to determine which intervals satisfy the inequality. Choose -1, 1, and 4 as test numbers. Substituting these numbers yields: \((-1)^{4}((-1)-3)\) is positive, \(1^{4}(1-3)\) is negative, and \(4^{4}(4-3)\) is positive. Hence, the intervals that satisfy the inequality are \(0\leq x\leq 3\). The inequality is nonstrict because it allows \(x^{4}(x-3)=0\).
04

Draw the graph

Sketch a graph with x-axis from -1 to 4. Mark \(x=0\) and \(x=3\). Draw a solid dot at these points, because the solution includes 0 and 3. Between these points draw a line, showing that all values on this interval are solutions. Do not draw lines outside of this interval, because the inequality is not satisfied there.

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Most popular questions from this chapter

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=5 x^{5}-10 x$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{2}-2 x+17$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}-3 x^{2}+4 x-2$$
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