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Chapter 2: Problem 39

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x^{3}+x^{2}+4 x+4>0$$

Short Answer

Expert verified
The solution is heavily dependent on the roots of the polynomial. There will be one or several intervals that satisfy the inequality, and these intervals can be represented in interval notation. The actual result will depend on the polynomial roots found in step 1.

Step by step solution

01

Set the Polynomial equal to Zero

Firstly, to find the critical points of the polynomial, set the polynomial equal to zero and solve for \(x\). i.e. solve for \(x\) in \(x^{3}+x^{2}+4 x+4 = 0\). This can be achieved by applying factoring or using the Rational Root Theorem. If the roots are not rational numbers, you may have to resort to numerical methods or use a graphing calculator for the roots.
02

Identify the Intervals

The critical points derived determine several intervals of \(x\). This will split the number line into several intervals. Once we find those points, we place them on a number line.
03

Test the Intervals

Now, choose test points within these intervals and substitute them back into the inequality. This will show whether the inequality is true or false within these intervals.
04

Solution set in Interval Notation

Lastly, express the solution set, the \(x\) values for which the inequality is true, in interval notation. Only the sections of the number line that make the original inequality true are included in the interval.

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