Worksheets · Precalculus

Precalculus synthetic division worksheet

In Precalculus synthetic division is how you find every zero of a polynomial. Divide out a zero you know, and the quotient is a smaller polynomial you can factor or solve, even when its zeros are complex.

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

Synthetic Division

Date Period

Divide using synthetic division.

  1. −6a3−13a2+17a+1a+3
  2. Use synthetic division to find p(−1) for p(n)=3n3−4n2−5n−3
  3. −5p3+25p2−24p+9p−4
  4. Use synthetic division to find p(1) for p(y)=−7y3−3y2−3y−7
  5. −5b3−17b2+6b−15b+4
  6. Use synthetic division to find p(4) for p(a)=−8a3−8a2+2a+8
  7. −3r3−15r2−18r−30r+4
  8. Use synthetic division to find p(−1) for p(v)=−9v3−2v2+8v+1
  9. a3−7a2+9a+7a−2
  10. Use synthetic division to find p(2) for p(b)=−b3+b2+9b+3
  11. −r3+3r2+3r−10r−1
  12. Use synthetic division to find p(−1) for p(b)=−4b3+7b2+8b+3
  13. −2b3+5b2+24b−3b−5
  14. Use synthetic division to find p(1) for p(m)=7m3−3m2−7m+5
  15. 2y3−5y2−4y+6y−3
  16. Use synthetic division to find p(2) for p(v)=4v3+v2−2v+1
  17. 3x3−19x2+21x−3x−5
  18. Use synthetic division to find p(3) for p(b)=−2b3−6b2+b+6
  19. 6x3+10x2+8x−4x+1
  20. Use synthetic division to find p(−1) for p(y)=−7y3−y2+6y−8

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

Sigma Prep · sigmaprep.io/worksheets

Name

Synthetic DivisionAnswer keyVersion 1

Date Period

Divide using synthetic division.

  1. −6a3−13a2+17a+1a+3−6a2+5a+2−5a+3
  2. Use synthetic division to find p(−1) for p(n)=3n3−4n2−5n−3−5
  3. −5p3+25p2−24p+9p−4−5p2+5p−4−7p−4
  4. Use synthetic division to find p(1) for p(y)=−7y3−3y2−3y−7−20
  5. −5b3−17b2+6b−15b+4−5b2+3b−6+9b+4
  6. Use synthetic division to find p(4) for p(a)=−8a3−8a2+2a+8−624
  7. −3r3−15r2−18r−30r+4−3r2−3r−6−6r+4
  8. Use synthetic division to find p(−1) for p(v)=−9v3−2v2+8v+10
  9. a3−7a2+9a+7a−2a2−5a−1+5a−2
  10. Use synthetic division to find p(2) for p(b)=−b3+b2+9b+317
  11. −r3+3r2+3r−10r−1−r2+2r+5−5r−1
  12. Use synthetic division to find p(−1) for p(b)=−4b3+7b2+8b+36
  13. −2b3+5b2+24b−3b−5−2b2−5b−1−8b−5
  14. Use synthetic division to find p(1) for p(m)=7m3−3m2−7m+52
  15. 2y3−5y2−4y+6y−32y2+y−1+3y−3
  16. Use synthetic division to find p(2) for p(v)=4v3+v2−2v+133
  17. 3x3−19x2+21x−3x−53x2−4x+1+2x−5
  18. Use synthetic division to find p(3) for p(b)=−2b3−6b2+b+6−99
  19. 6x3+10x2+8x−4x+16x2+4x+4−8x+1
  20. Use synthetic division to find p(−1) for p(y)=−7y3−y2+6y−8−8

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

How to do these

Find all zeros of p(x)=x3−3x2+4x−12, given x=3 is a zero

  1. Divide by x - 3: the bottom row is 1, 0, 4, 0, so the remainder is 0.
  2. The quotient is x² + 4.
  3. x² + 4 = 0 gives x = ±2i.

The answer is x=3, 2i, −2i.

Where students go wrong

Dividing by 2x - 1 with 1 in the box. Use 1/2, then divide the quotient by 2, since 2x - 1 is 2(x - 1/2).

Also called dividing polynomials by x - c.

What you can put on this worksheet

Divide by x - c
−6v2+14v+12v−3 → −6v−4
Evaluate p(c) by synthetic substitution
Use synthetic division to find p(3) for p(v)=2v2−3v+5 → 14
Missing terms, quotient and remainder
7m5−13m4+7m3−9m2+11m−8m−1 → 7m4−6m3+m2−8m+3−5m−1

Questions about these worksheets

Yes. Every synthetic division 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 synthetic division 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 3: divide by x - c, evaluate p(c) by synthetic substitution and missing terms, quotient and remainder. Tick as many as you want and set how many of each, or let it spread them evenly.

Putting the wrong number in the box. Dividing by x - 2 means 2 goes in, and dividing by x + 3 means -3 does, because the divisor is x minus the number. The other slip is a missing term: if the dividend has no x term you still write a 0 for it, or every column after it shifts and the whole answer is wrong.