2.3: Solution of Polynomial Inequalities by Graphing
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In this section, we will combine the concepts of the previous two sections to solve polynomial inequalities. In Section 2.2, we solved equations by graphing and finding the x -values which made y=0. In solving an inequality, we will be concerned with finding the range of x values that make y either greater than or less than 0, depending on the given problem.
Example
Solve the given inequality.
2x3+8x2+5x−3≥0
First, we graph the function:
Then we identify the intervals of x -values that make the y value greater than or equal to zero, as indicated in the problem.
The indicated roots of the function (A,B and C) are the x -values that make y equal to zero. These points divide the graph between the regions where y is greater than zero and the regions where y is less than zero. The solution to the given inequlaity 2x3+8x2+5x−3≥0 are A≤x≤B OR x≥C
When we find the values of A,B and C:A=−3,B≈−1.366 and C≈0.366, we can
complete the solution to the problem.
2x3+8x2+5x−3≥0
−3≤x≤−1.366 OR x≥0.366
Example
Solve the given inequality.
x4−2x3−5x2+8x+3≤0
First, we graph the function:
In this problem, we're looking for the intervals of x values that make y less than or equal to zero. First, we identify the roots of the function:
Next, we'll identify the intervals where the y values are less than zero:
So, the solution to the original inequality is:
x4−2x3−5x2+8x+3≤0
−2.034≤x≤−0.320 OR 1.806≤x≤2.549
In the next example we'll be looking to identify both the intervals where y is greater than zero, and the intervals where y is less than zero.
Example
Determine the interval(s) for which x3+5x2+5x+1≥0
Determine the interval(s) for which x3+5x2+5x+1<0
Once again, we'll start by graphing the function to find the roots:
Now that we've indentified the roots, we can determine where the y values are greater than zero and where they're less than zero.
For y≥0, we can see that this corresponds to: −3.732≤x≤−1 OR x≥−0.268
For y<0, we can see that this corresponds to: x<−3.732OR−1<x<−0.268
Exercises 2.3
1) Determine the interval(s) for which x3−4x2+2x+3≥0
Determine the interval(s) for which x3−4x2+2x+3<0
2) Determine the interval(s) for which 4x3−4x2−19x+10≥0
Determine the interval(s) for which 4x3−4x2−19x+10<0
3) Determine the interval(s) for which x3−2.5x2−7x−1.5≥0
Determine the interval(s) for which x3−2.5x2−7x−1.5<0
4) Determine the interval(s) for which x3−3.5x2+0.5x+5≥0
Determine the interval(s) for which x3−3.5x2+0.5x+5<0
5) Determine the interval(s) for which 6x4−13x3+2x2−4x+15≥0
Determine the interval(s) for which 6x4−13x3+2x2−4x+15<0
6) Determine the interval(s) for which x4−x3−x2+3x−5≥0
Determine the interval(s) for which x4−x3−x2+3x−5<0
7) Determine the interval(s) for which 3x4+3x3−14x2−x+3≥0
Determine the interval(s) for which 3x4+3x3−14x2−x+3<0
8) Determine the interval(s) for which 4x4−4x3−7x2+4x+3≥0
Determine the interval(s) for which 4x4−4x3−7x2+4x+3<0
Determine the interval(s) that satisfy each inequality.
9) x3+x2−5x+3≤0
10) x3−7x+6>0
11) x3−13x+12>0
12) x4−10x2+9<0
13) 6x4−9x2−4x+12≥0
14) x4−5x3+20x−16>0
15) x3−2x2−7x+6≤0
16) x4−6x3+2x2−5x+2≤0
17) 2x4+3x3−2x2−4x+2>0
18) x5+5x4−4x3+3x2−2≤0