Solving Polynomial Inequalities
We've talked a lot about how to solve polynomial equations, but we've given considerable less thought to solving polynomial inequalities. There's not much of a difference, but there are a couple of extra steps we will consider.
How to solve Polynomial Inequalities
Step 1: Rearrange the inequality such that one side of the inequality is 0 and all variables and constants are on the opposite side.
Step 2: Factor the polynomial side of the inequality.
Step 3: Replace the inequality symbol with an equals sign and find the zeros of the function.
Step 4: On a number line, mark each of the zeros of the function.
Step 5: Let the zeros of the function divide the number line into segments. Select one value from each of the intervals on the number line created by the zeros.
Step 6: Evaluate the inequality from Step 1 at each of the points selected from Step 5. If the inequality is true, that interval of the number line from which the evaluated value was selected is a solution. If the inequality is false, disregard that interval of the number line.
A polynomial inequality is a mathematical statement that relates a polynomial expression as either less than or greater than another. We can use sign charts to solve polynomial inequalities with one variable.
Graphs are helpful in providing a visualization to the solutions of polynomial inequalities. Examine the graph below to see the relationship between a graph of a polynomial and its corresponding sign chart.
It is now time to look at solving some more difficult inequalities. In this section we will be solving (single) inequalities that involve polynomials of degree at least two. Or, to put it in other words, the polynomials won&rsquo