Polynomial Inequalities Practice Problems

Q: Why can't I solve a polynomial inequality by isolating x?

Isolating x would require dividing by variable factors whose sign is unknown, and dividing an inequality by a negative flips it. The sign-chart method avoids this by analyzing the sign of one factored expression versus zero.

Q: Are the critical values included?

For , no (parentheses). For ≤ or ≥, yes (brackets) — the polynomial equals zero there, which satisfies a non-strict inequality.

Q: What happens at a repeated factor like (x − 2)²?

The polynomial touches zero at x = 2 but does not change sign, because a squared factor is never negative.

Q: Does my interval have to use the ∪ symbol?

No. A plain U between the two pieces works just as well as ∪, and you can type the inequality form instead if you prefer. Include the critical values as square brackets when the inequality is ≤ or ≥, and leave them as parentheses when it is strict.

Polynomial Inequalities Practice Problems

The Polynomial Inequalities lesson lays out the sign-chart method step by step; this set is 12 to practice on, each fully solved.

How to use this set

Move all terms to one side, factor, list the critical values, test one point per interval, and select the intervals whose sign matches — adding the critical values for / . Type the interval into the answer box and press Check Answers. (-inf, -2] U [5, inf) and x <= -2 or x >= 5 are both accepted, so use whichever notation you are practising.

Common mistakes to watch for

Dividing by a variable factor. is not ; move everything over and factor.

Not setting one side to zero first.

Forgetting the endpoints for / , or including them for / .

Assuming the sign always alternates — it doesn’t at a squared factor.

Where to go next

Finding the critical values uses Factoring Quadratics. The next step, with denominators, is the Rational Inequalities Practice Problems. The answer format is Interval Notation. For the full method, see the Polynomial Inequalities lesson.

Frequently Asked Questions

Why can't I solve a polynomial inequality by isolating x?+

Isolating would require dividing by variable factors whose sign is unknown, and dividing an inequality by a negative flips it. The sign-chart method avoids this by analyzing the sign of one factored expression versus zero.

What are critical values?+

The roots of the factored polynomial — the only places the polynomial can change sign. They split the number line into test intervals.

How do I test an interval?+

Pick any number strictly inside it, substitute into the factored form, and record whether the result is positive or negative. One test point settles the whole interval.

Are the critical values included?+

For or , no (parentheses). For or , yes (brackets) — the polynomial equals zero there, which satisfies a non-strict inequality.

What happens at a repeated factor like (x − 2)²?+

The polynomial touches zero at but does not change sign, because a squared factor is never negative.

Does my interval have to use the ∪ symbol?+

No. A plain U between the two pieces works just as well as ∪, and you can type the inequality form instead if you prefer. Include the critical values as square brackets when the inequality is ≤ or ≥, and leave them as parentheses when it is strict.

>0Polynomial Inequalities Practice Problems

Polynomial Inequalities Practice Problems

How to use this set

Common mistakes to watch for

Where to go next

Frequently Asked Questions

More practice problems

Polynomial Inequalities Practice Problems