Worksheets · Algebra 2

Rational root theorem worksheet

Every polynomial here has one zero that is a fraction, which is what the rational root theorem is for: the top divides the constant, the bottom divides the leading coefficient. That gives a short list to test.

Select your difficulty

Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Rational Zeros

Date Period

Find all zeros.

  1. 2a3−13a2−15a+126=0
  2. 4a3+11a2+7a=0
  3. 3n3+20n2+12n=0
  4. 4m3−17m2+m+12=0
  5. 2b3−7b2−18b+63=0
  6. 2n3+17n2+27n−18=0
  7. 4a3−a2−26a−21=0
  8. 4m3−39m2+116m−105=0
  9. 4k3−7k2−62k−15=0
  10. 3k3+k2−34k+40=0
  11. 2y3+7y2+7y+2=0
  12. 3a3−28a2+69a−20=0
  13. 2b3+23b2+86b+105=0
  14. 3a3+34a2+101a+30=0
  15. 4m3+15m2−4m=0
  16. 2r3−5r2−67r+210=0
  17. 2n3−19n2+54n−45=0
  18. 2x3+x2−41x−70=0
  19. 3k3+13k2+8k−12=0
  20. 2b3+5b2−26b−56=0

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

Sigma Prep · sigmaprep.io/worksheets

Name

Rational ZerosAnswer keyVersion 1

Date Period

Find all zeros.

  1. 2a3−13a2−15a+126=0{72,6,−3}
  2. 4a3+11a2+7a=0{−74,−1,0}
  3. 3n3+20n2+12n=0{−23,0,−6}
  4. 4m3−17m2+m+12=0{−34,1,4}
  5. 2b3−7b2−18b+63=0{72,−3,3}
  6. 2n3+17n2+27n−18=0{12,−3,−6}
  7. 4a3−a2−26a−21=0{−74,3,−1}
  8. 4m3−39m2+116m−105=0{74,3,5}
  9. 4k3−7k2−62k−15=0{−14,5,−3}
  10. 3k3+k2−34k+40=0{53,−4,2}
  11. 2y3+7y2+7y+2=0{−12,−1,−2}
  12. 3a3−28a2+69a−20=0{13,4,5}
  13. 2b3+23b2+86b+105=0{−72,−3,−5}
  14. 3a3+34a2+101a+30=0{−13,−5,−6}
  15. 4m3+15m2−4m=0{14,0,−4}
  16. 2r3−5r2−67r+210=0{72,5,−6}
  17. 2n3−19n2+54n−45=0{32,3,5}
  18. 2x3+x2−41x−70=0{−72,−2,5}
  19. 3k3+13k2+8k−12=0{23,−2,−3}
  20. 2b3+5b2−26b−56=0{72,−4,−2}

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

How to do these

2x2−x−6=0

  1. The top divides 6, the bottom divides 2.
  2. So the candidates are plus or minus 1, 2, 3, 6, 1/2 and 3/2.
  3. Testing finds -3/2 and 2.

The answer is x=−32 or x=2.

Where students go wrong

Only testing whole numbers. If the leading coefficient is not 1 then a fractional zero is likely, and a student who tries 1, 2 and 3 and gives up concludes wrongly that it does not factor.

Also called rational zero theorem, finding rational zeros or possible rational roots.

Questions about these worksheets

Yes. Every rational root theorem 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 rational root theorem 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.

Only testing whole numbers. If the leading coefficient is not 1 then a fractional zero is likely, and a student who tries 1, 2 and 3 and gives up concludes wrongly that it does not factor.