Worksheets · Algebra 2

Factors and zeros worksheet

A zero and a factor are the same fact written two ways: if 3 is a zero then x - 3 is a factor. Both directions are on this sheet, because students who can go one way often cannot go the other.

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

Factors and Zeros

Date Period

Answer each question.

  1. Write a polynomial with zeros −7,0,7
  2. y3−2y2−9y+18=0, given that y+3 is a factor
  3. Write a polynomial with zeros 7,−1,0
  4. n3+10n2+29n+20=0, given that n+1 is a factor
  5. Write a polynomial with zeros −6,−2,4
  6. b3+13b2+47b+35=0, given that b+5 is a factor
  7. Write a polynomial with zeros 4,−3,−4
  8. a3−21a+20=0, given that a−1 is a factor
  9. Write a polynomial with zeros −1,4,7
  10. b3+4b2−17b−60=0, given that b−4 is a factor
  11. Write a polynomial with zeros −4,−5,3
  12. v3+17v2+94v+168=0, given that v+6 is a factor
  13. Write a polynomial with zeros 5,−4,7
  14. r3+r2−9r−9=0, given that r+1 is a factor
  15. Write a polynomial with zeros 6,3,5
  16. y3+7y2−36y−252=0, given that y+6 is a factor
  17. Write a polynomial with zeros 6,0,7
  18. r3−5r2−29r+105=0, given that r+5 is a factor
  19. Write a polynomial with zeros 7,−5,−2
  20. n3+2n2−5n−6=0, given that n+1 is a factor

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

Sigma Prep · sigmaprep.io/worksheets

Name

Factors and ZerosAnswer keyVersion 1

Date Period

Answer each question.

  1. Write a polynomial with zeros −7,0,7x3−49x
  2. y3−2y2−9y+18=0, given that y+3 is a factor{−3,2,3}
  3. Write a polynomial with zeros 7,−1,0x3−6x2−7x
  4. n3+10n2+29n+20=0, given that n+1 is a factor{−1,−4,−5}
  5. Write a polynomial with zeros −6,−2,4x3+4x2−20x−48
  6. b3+13b2+47b+35=0, given that b+5 is a factor{−5,−7,−1}
  7. Write a polynomial with zeros 4,−3,−4x3+3x2−16x−48
  8. a3−21a+20=0, given that a−1 is a factor{1,4,−5}
  9. Write a polynomial with zeros −1,4,7x3−10x2+17x+28
  10. b3+4b2−17b−60=0, given that b−4 is a factor{4,−5,−3}
  11. Write a polynomial with zeros −4,−5,3x3+6x2−7x−60
  12. v3+17v2+94v+168=0, given that v+6 is a factor{−6,−4,−7}
  13. Write a polynomial with zeros 5,−4,7x3−8x2−13x+140
  14. r3+r2−9r−9=0, given that r+1 is a factor{−1,−3,3}
  15. Write a polynomial with zeros 6,3,5x3−14x2+63x−90
  16. y3+7y2−36y−252=0, given that y+6 is a factor{−6,−7,6}
  17. Write a polynomial with zeros 6,0,7x3−13x2+42x
  18. r3−5r2−29r+105=0, given that r+5 is a factor{−5,3,7}
  19. Write a polynomial with zeros 7,−5,−2x3−39x−70
  20. n3+2n2−5n−6=0, given that n+1 is a factor{−1,−3,2}

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

How to do these

Write a polynomial with zeros 2,−3,5

  1. Each zero gives a factor: (x - 2), (x + 3), (x - 5).
  2. Multiply them out.

The answer is x3−4x2−11x+30.

Where students go wrong

Copying the sign of the zero into the factor. A zero of -3 gives the factor (x + 3), not (x - 3), because it is the value that makes the parenthesis zero. Getting that backwards flips the whole polynomial.

Also called roots of a polynomial, zeros of a function or the factor theorem.

What you can put on this worksheet

Write a polynomial from its zeros
Write a polynomial with zeros 1,3 → x2−4x+3
One factor given, find the other zeros
v3+v2−17v+15=0, given that v−1 is a factor → {1,3,−5}

Questions about these worksheets

Yes. Every factors and zeros 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 factors and zeros 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 2: write a polynomial from its zeros and one factor given, find the other zeros. Tick as many as you want and set how many of each, or let it spread them evenly.

Copying the sign of the zero into the factor. A zero of -3 gives the factor (x + 3), not (x - 3), because it is the value that makes the parenthesis zero. Getting that backwards flips the whole polynomial.