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Chapter 3: Problem 16

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. $$ 3 x^{2}+16 x<-5 $$

Short Answer

Expert verified
TBD- The short answer will be dependent on the specific solutions obtained for the inequality, and will be the interval notation of these solutions.

Step by step solution

01

Rewrite Inequality as Quadratic Equation

Start by rewriting the inequality in the standard form of a quadratic equation. Do this by moving all terms to one side, resulting in \(3x^2 + 16x + 5 > 0\).
02

Solve Quadratic Equation

Now solve the associated quadratic equation, \(3x^2 + 16x + 5 = 0\) which will give the possible solutions. In this case, it is best to use the Quadratic Formula which is \(x = [-b 卤 sqrt(b^2 - 4ac)] / 2a\). Plug in \(a=3\), \(b=16\), and \(c=5\) and solve for the two possible values of \(x\).
03

Test Solutions on the Inequality

Test the solutions (interval endpoints) on the inequality. The intervals are divided by the x-values obtained, from negative infinity to x1, then from x1 to x2, and finally, from x2 to positive infinity. For each interval, pick a number within it to verify which intervals satisfy the inequality.
04

Write Solution in Interval Notation and Graph

The solution to the inequality is the union of the intervals which made the inequality hold true. Write the solution in interval notation. On the number line, the solution will be represented by a line segment or ray for each interval.

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Most popular questions from this chapter

Can the graph of a polynomial function have no \(y\) -intercept? Explain.

There is more than one third-degree polynomial function with the same three x-intercepts.

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. $$f(x)=-2 x^{3}(x-1)^{2}(x+5)$$

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. $$f(x)=6 x^{3}-9 x-x^{5}$$

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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