Polynomial Inequalities Worksheet: Sign Charts + Answer Key
Practice polynomial inequalities with sign charts, factoring, interval notation, and graph interpretation. Includes 18 problems, three difficulty levels, and a complete answer key.
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Free to use for individual classroom, tutoring, homework, and review practice. Last reviewed for mathematical accuracy: August 29, 2026.
Polynomial Inequalities Sign-Chart Cheat Sheet
Use this quick reference while solving the worksheet. The same cheat sheet is available as a one-page printable PDF.
Polynomial Inequality Sign Chart Example
Consider the expression below. Its three zeros divide the number line into four intervals.
Zeros: x = -3, 1, 4
| Interval | (-∞, -3) | (-3, 1) | (1, 4) | (4, ∞) |
|---|---|---|---|---|
| Sign | Negative | Positive | Negative | Positive |
| Use if p(x) ≥ 0? | No | Yes | No | Yes |
Because each zero in this example has odd multiplicity, the sign changes every time the graph crosses a zero. For (x + 3)(x - 1)(x - 4) ≥ 0, choose the positive intervals and include the zeros because equality is allowed.
Important Multiplicity Rule
At a zero with odd multiplicity, the polynomial changes sign. At a zero with even multiplicity, the polynomial touches the x-axis and keeps the same sign on both sides. This is why a repeated factor such as (x - 2)² needs special attention on a sign chart.
Worked Polynomial Inequality Examples
These three examples show the progression students see in the worksheet: an inequality already in factored form, an inequality that must be factored first, and a repeated zero with even multiplicity.
Example 1: Already Factored
The zeros are -2 and 5, creating the intervals (-∞, -2), (-2, 5), and (5, ∞).
The signs are positive, negative, positive, so choose the two positive intervals.
Answer: (-∞, -2) ∪ (5, ∞)
Example 2: Factor First
Factor the quadratic as (x - 2)(x - 3). The zeros are 2 and 3.
The quadratic is negative between its two zeros. Equality is allowed, so both endpoints are included.
Answer: [2, 3]
Example 3: Repeated Zero
The zeros are -1 and 4. The zero at -1 has even multiplicity, so the sign does not change there.
The expression is negative on both sides of -1 until x = 4. Because the inequality is strict, the zeros are excluded.
Answer: (-∞, -1) ∪ (-1, 4)
How to Write Polynomial Inequality Answers in Interval Notation
For a strict inequality such as p(x) > 0 or p(x) < 0, zeros are excluded and written with parentheses. For p(x) ≥ 0 or p(x) ≤ 0, zeros that satisfy the equality are included and written with brackets. Infinity is never included, so always use a parenthesis next to ∞ or -∞.
| Condition | Endpoint Rule |
|---|---|
| p(x) > 0 | Zeros are excluded; use parentheses. |
| p(x) < 0 | Zeros are excluded; use parentheses. |
| p(x) ≥ 0 | Zeros that satisfy equality are included; use brackets. |
| p(x) ≤ 0 | Zeros that satisfy equality are included; use brackets. |
Polynomial Inequalities Worksheet Questions
The worksheet contains 18 problems arranged in three levels so students can move from basic sign charts to factoring, repeated roots, higher-degree polynomials, and graph interpretation.
Directions: Solve each inequality. Show factoring when needed, mark the zeros on a number line or sign chart, determine the sign on each interval, and write the final solution in interval notation. Use set notation when an isolated point is part of the solution.
Level 1: Build the Sign Chart
These expressions are already factored. Focus on zeros, interval signs, endpoint inclusion, and multiplicity.
Level 2: Factor First
Factor each polynomial completely before building the sign chart.
Level 3: Multiplicity & Higher Degree
These problems combine repeated zeros, higher-degree factoring, and more complex sign patterns.
Optional Extension
For any two problems above, sketch a possible polynomial graph that matches your sign chart. Show the correct x-intercepts and whether the graph crosses or touches the x-axis at repeated zeros.
Polynomial Inequalities Worksheet Answer Key
These are the final solutions. The downloadable answer-key PDF includes additional factoring and multiplicity notes.
| # | Factoring / Interpretation | Solution |
|---|---|---|
| 1 | (x - 2)(x + 5) > 0 | (-∞, -5) ∪ (2, ∞) |
| 2 | (x + 1)(x - 4) ≤ 0 | [-1, 4] |
| 3 | (x - 3)(x + 2)(x + 6) ≥ 0 | [-6, -2] ∪ [3, ∞) |
| 4 | (x + 4)(x - 1)(x - 5) < 0 | (-∞, -4) ∪ (1, 5) |
| 5 | (x - 2)²(x + 3) ≥ 0 | [-3, ∞) |
| 6 | Zeros -4 and 2; graph opens upward | [-4, 2] |
| 7 | x² - 7x + 10 = (x - 2)(x - 5) | (-∞, 2) ∪ (5, ∞) |
| 8 | x² + x - 12 = (x + 4)(x - 3) | [-4, 3] |
| 9 | x³ - 4x² - x + 4 = (x - 4)(x - 1)(x + 1) | [-1, 1] ∪ [4, ∞) |
| 10 | x³ + x² - 9x - 9 = (x - 3)(x + 1)(x + 3) | (-∞, -3) ∪ (-1, 3) |
| 11 | 2x² - 5x - 3 = (2x + 1)(x - 3) | (-∞, -1/2] ∪ [3, ∞) |
| 12 | Use the intervals where the graph is below the x-axis | (-∞, -3) ∪ (1, 4) |
| 13 | (x - 3)²(x + 1)(x - 5) > 0 | (-∞, -1) ∪ (5, ∞) |
| 14 | (x + 2)²(x - 1)(x - 4) ≤ 0 | {-2} ∪ [1, 4] |
| 15 | (x² - 9)(x - 2) = (x + 3)(x - 3)(x - 2) | [-3, 2] ∪ [3, ∞) |
| 16 | x⁴ - 5x² + 4 = (x - 2)(x - 1)(x + 1)(x + 2) | (-∞, -2] ∪ [-1, 1] ∪ [2, ∞) |
| 17 | x⁴ + x³ - 7x² - x + 6 = (x - 2)(x - 1)(x + 1)(x + 3) | (-3, -1) ∪ (1, 2) |
| 18 | Touch at -2; cross at 1 and 5 | (-∞, 1] ∪ [5, ∞) |
Common Polynomial Inequality Mistakes
Most errors happen after the factoring step. The worksheet deliberately gives students repeated opportunities to connect zeros with sign changes and endpoint rules.
Teacher and Tutor Use
Level 1 works well immediately after students learn sign charts. Level 2 adds factoring and is appropriate for a standard Algebra 2 lesson or review. Level 3 is better for advanced Algebra 2 or precalculus because students must track repeated zeros and higher-degree sign changes.
For a shorter assignment, use Problems 1-4, 7-10, and 13-16. For additional challenge, require students to justify each interval sign without substituting a test value, then sketch a possible polynomial graph that matches the sign chart.
Related Polynomial Worksheets and Activities
Students who need more practice with factoring, polynomial structure, or multiple representations can continue with these related resources.
Additional Free Learning References
For students who want another explanation or extra practice, these free nonprofit education resources reinforce the same skills used on this worksheet.
Created and reviewed by Berke Sahbazoglu
Owner and lead math tutor, Burke Tutoring in Fremont. 10+ years tutoring Algebra I & II. This worksheet was reviewed for factorization, sign changes, endpoint inclusion, interval notation, and repeated-root behavior.
Last updated August 29, 2026. The downloadable student worksheet and answer key match the examples and difficulty progression shown on this page.