Worksheets · Precalculus

Polynomial inequalities worksheet for Precalculus

The zeros split the number line into pieces, and the sign can only change at a zero. Test one point in each piece, or track the sign across the chart, and collect the pieces that work.

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

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

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

Sigma Prep · sigmaprep.io/worksheets

Name

Polynomial InequalitiesAnswer keyVersion 1

Date Period

Solve each inequality. Write each answer in interval notation.

  1. Solve polynomial inequalities

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

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

How to do these

(x−2)(x+3)(x−5)>0

  1. The zeros are -3, 2 and 5.
  2. Far right is positive, and the sign flips at each zero.
  3. Keep the positive pieces.

The answer is (−3,2)∪(5,∞).

Where students go wrong

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.

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.

Precalculus is usually taken around 11th or 12th 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.