Worksheets · Precalculus

Solving polynomial equations worksheet for Precalculus

Precalculus solves fourth-degree equations for every root. Take out any common factor first, test the rational candidates, divide out each one that works with synthetic division, and finish by factoring the quadratic that is left.

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.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Solving Polynomial Equations

Date Period

Solve each equation.

  1. r4−31r2−42r+72=0
  2. p4+9p3−11p2−225p−350=0
  3. k4+4k3−17k2−24k+36=0
  4. 4x4+44x3−76x2−2156x−5880=0
  5. 2p4−8p3−78p2+92p+160=0
  6. 4v4+8v3−396v2−512v+8960=0
  7. 4x4−24x3−124x2+480x=0
  8. y4−6y3−57y2+286y+336=0
  9. y4−52y2+96y=0
  10. 2p4−16p3−30p2+388p−560=0
  11. m4−3m3−16m2−12m=0
  12. b4+12b3+26b2−108b−315=0
  13. 4v4−36v3+56v2+96v=0
  14. 2p4+10p3−2p2−10p=0
  15. 4y4+36y3+32y2−240y=0
  16. x4−7x3+5x2+31x−30=0
  17. m4−5m3−54m2+344m−448=0
  18. 3n4+12n3−87n2+72n=0
  19. 3k4−15k3+15k2+15k−18=0
  20. y4−7y2−6y=0

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

Sigma Prep · sigmaprep.io/worksheets

Name

Solving Polynomial EquationsAnswer keyVersion 2

Date Period

Solve each equation.

  1. r4−31r2−42r+72=0{−4,−3,1,6}
  2. p4+9p3−11p2−225p−350=0{−7,−5,−2,5}
  3. k4+4k3−17k2−24k+36=0{−6,−2,1,3}
  4. 4x4+44x3−76x2−2156x−5880=0{−7,−6,−5,7}
  5. 2p4−8p3−78p2+92p+160=0{−5,−1,2,8}
  6. 4v4+8v3−396v2−512v+8960=0{−8,−7,5,8}
  7. 4x4−24x3−124x2+480x=0{−5,0,3,8}
  8. y4−6y3−57y2+286y+336=0{−7,−1,6,8}
  9. y4−52y2+96y=0{−8,0,2,6}
  10. 2p4−16p3−30p2+388p−560=0{−5,2,4,7}
  11. m4−3m3−16m2−12m=0{−2,−1,0,6}
  12. b4+12b3+26b2−108b−315=0{−7,−5,−3,3}
  13. 4v4−36v3+56v2+96v=0{−1,0,4,6}
  14. 2p4+10p3−2p2−10p=0{−5,−1,0,1}
  15. 4y4+36y3+32y2−240y=0{−6,−5,0,2}
  16. x4−7x3+5x2+31x−30=0{−2,1,3,5}
  17. m4−5m3−54m2+344m−448=0{−8,2,4,7}
  18. 3n4+12n3−87n2+72n=0{−8,0,1,3}
  19. 3k4−15k3+15k2+15k−18=0{−1,1,2,3}
  20. y4−7y2−6y=0{−2,−1,0,3}

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

How to do these

x4−x3−7x2+x+6=0

  1. Test factors of 6: x = 1 gives 1 - 1 - 7 + 1 + 6 = 0, so x - 1 is a factor.
  2. Synthetic division by 1 leaves x³ - 7x - 6, and x = -1 makes that 0.
  3. Dividing by x + 1 leaves x² - x - 6 = (x - 3)(x + 2).

The answer is {1,−1,3,−2}.

Where students go wrong

Stopping after the first root. Once x - 1 divides out, the cubic that is left still holds three roots; keep testing candidates until only a quadratic remains.

Also called solving cubic equations, higher degree equations or finding all real zeros.

What you can put on this worksheet

Cubics and quartics that factor
v3+v2−10v+8=0 → {−4,1,2}
Equations with imaginary roots
x4−81=0 → {3,−3,3i,−3i}
Quadratic form with imaginary or irrational roots, degree 5 to 8
k4−28k2+75=0 → {−5,−3,3,5}
Quartics like 3x^4 - 13x^2 + 4
Find all zeros: 4x4−41x2+72=0 → {32, −32, 22, −22}
Quintics with a leading coefficient
Find all zeros:
4x5+5x4−52x3−65x2+144x+180=0 → {−54, 2, −2, 3, −3}

Questions about these worksheets

Yes. Every solving polynomial equations 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 solving polynomial equations 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 5: cubics and quartics that factor, equations with imaginary roots, quadratic form with imaginary or irrational roots, degree 5 to 8, quartics like 3x^4 - 13x^2 + 4 and quintics with a leading coefficient. Tick as many as you want and set how many of each, or let it spread them evenly.

Stopping at one root. A cubic has three roots and a quartic has four, so finding the first factor is the start of the job, not the end. Divide it out and keep going.