Worksheets · Algebra 2

Polynomial inequalities worksheet for Algebra 2

Algebra 2 solves polynomial inequalities with a sign chart built from factored form and writes the answer in interval notation, often as a union. Watch for repeated factors and for ≤ or ≥, where the zeros belong in the answer and can even appear as single points.

Also taught in Precalculus, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polynomial Inequalities

Date Period

Solve each inequality. Write each answer in interval notation.

  1. Solve polynomial inequalities

    (x−5)(x−3)(x+4)<0
  2. Solve: x3>25x
  3. (x+1)(x+5)(x+7)<0
  4. Solve: (x+2)2(x+3)<0
  5. −(x−4)(x−1)(x+7)<0
  6. Solve: −(x−1)2(x−5)>0
  7. (x−4)(x+6)(x+7)≤0
  8. Solve: x4>16x2
  9. (x−5)(x−4)(x+5)≤0
  10. Solve: −(x+2)2(x−3)>0
  11. (x−6)(x−1)(x+1)≥0
  12. Solve: (x−5)2(x+3)≤0
  13. (x−7)(x−6)(x+4)>0
  14. Solve: (x+5)2(x+2)>0
  15. −(x−6)(x−5)(x+7)>0
  16. Solve: (x+4)2(x+3)≤0
  17. (x−3)(x+4)(x+5)≥0
  18. Solve: (x+3)2(x−4)≥0
  19. (x−5)(x+4)(x+7)>0
  20. Solve: (x−4)2(x+2)>0

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polynomial InequalitiesAnswer keyVersion 2

Date Period

Solve each inequality. Write each answer in interval notation.

  1. Solve polynomial inequalities

    (x−5)(x−3)(x+4)<0(−∞,−4)∪(3,5)
  2. Solve: x3>25x(−5,0)∪(5,∞)
  3. (x+1)(x+5)(x+7)<0(−∞,−7)∪(−5,−1)
  4. Solve: (x+2)2(x+3)<0(−∞,−3)
  5. −(x−4)(x−1)(x+7)<0(−7,1)∪(4,∞)
  6. Solve: −(x−1)2(x−5)>0(−∞,1)∪(1,5)
  7. (x−4)(x+6)(x+7)≤0(−∞,−7]∪[−6,4]
  8. Solve: x4>16x2(−∞,−4)∪(4,∞)
  9. (x−5)(x−4)(x+5)≤0(−∞,−5]∪[4,5]
  10. Solve: −(x+2)2(x−3)>0(−∞,−2)∪(−2,3)
  11. (x−6)(x−1)(x+1)≥0[−1,1]∪[6,∞)
  12. Solve: (x−5)2(x+3)≤0(−∞,−3]∪{5}
  13. (x−7)(x−6)(x+4)>0(−4,6)∪(7,∞)
  14. Solve: (x+5)2(x+2)>0(−2,∞)
  15. −(x−6)(x−5)(x+7)>0(−∞,−7)∪(5,6)
  16. Solve: (x+4)2(x+3)≤0(−∞,−3]
  17. (x−3)(x+4)(x+5)≥0[−5,−4]∪[3,∞)
  18. Solve: (x+3)2(x−4)≥0{−3}∪[4,∞)
  19. (x−5)(x+4)(x+7)>0(−7,−4)∪(5,∞)
  20. Solve: (x−4)2(x+2)>0(−2,4)∪(4,∞)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

x3−4x≥0

  1. Factor: x(x - 2)(x + 2) ≥ 0, with zeros -2, 0 and 2.
  2. Test x = -1: (-1)(-3)(1) = 3, positive, and the sign flips at each single zero.
  3. Keep the positive pieces and the zeros.

The answer is [−2,0]∪[2,∞).

Where students go wrong

Dividing both sides by x to simplify. x can be negative or zero, so dividing by it can flip the inequality or lose solutions; factor it out and put it on the sign chart instead.

Also called sign chart or solving polynomial inequalities.

What you can put on this worksheet

Polynomial inequalities
(x−3)(x−1)>0 → (−∞,1)∪(3,∞)
Both sides, or a repeated root
Solve: x3≤16x → (−∞,−4]∪[0,4]
A leading coefficient other than 1
(4x−7)(x+5)>0 → (−∞,−5)∪(74,∞)
Perfect squares
x2−8x+16≤0 → {4}
Four factors
x(x+5)(x−1)(x−3)>0 → (−∞,−5)∪(0,1)∪(3,∞)
Write one from its solution set
Write a polynomial inequality whose solution set is [0,1] → One answer: x(x−1)≤0

Questions about these worksheets

Yes. Every polynomial inequalities sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the polynomial inequalities questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: polynomial inequalities, both sides, or a repeated root, a leading coefficient other than 1, perfect squares, four factors and write one from its solution set. Tick as many as you want and set how many of each, or let it spread them evenly.

Flipping the sign at every zero without checking the power. A squared factor touches zero and comes back the same sign, so the chart does not change there.