Worksheets · Precalculus

Descartes' rule of signs worksheet for Precalculus

Precalculus pairs Descartes' rule of signs with the rational root theorem to narrow down a polynomial's zeros before testing any. Count the sign changes in f(x) for the positive zeros and in f(-x) for the negative ones; each count can drop by an even number.

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

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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

Descartes' Rule of Signs

Date Period

Answer each question about the zeros of the polynomial.

  1. Find the possible positive real zeros

    Possible numbers of positive real zeros:
    f(x)=2x5+2x4−4x3−5x+3
  2. Possible numbers of positive real zeros:
    f(x)=−9x5+8x4−8x3−7x2+1
  3. Possible numbers of positive real zeros:
    f(x)=−4x5−7x4−8x3−4x+2
  4. Possible numbers of positive real zeros:
    f(x)=−2x5+5x3−3x2−8x−5
  5. Possible numbers of positive real zeros:
    f(x)=7x5−x4−3x2−9x−2
  6. Possible numbers of positive real zeros:
    f(x)=−8x5−x4+2x2−5x+1
  7. Possible numbers of positive real zeros:
    f(x)=−2x5−9x4−x3−4x2+1

  1. Find the possible negative real zeros

    Possible numbers of negative real zeros:
    f(x)=−x5−9x3−9x2−7x+6
  2. Possible numbers of negative real zeros:
    f(x)=9x5+3x4+8x3+6x+9
  3. Possible numbers of negative real zeros:
    f(x)=−3x5+6x4−7x2+3x+5
  4. Possible numbers of negative real zeros:
    f(x)=−3x5−8x4−8x2−2x+5
  5. Possible numbers of negative real zeros:
    f(x)=−4x5+2x3+7x2+x+4
  6. Possible numbers of negative real zeros:
    f(x)=3x5−7x4−6x3−5x2+6
  7. Possible numbers of negative real zeros:
    f(x)=6x5+x4+8x3+3x2−6

  1. Find possible positive and negative real zeros

    Possible numbers of positive and negative real zeros:
    f(x)=8x5−x4−5x3−4x2−6
  2. Possible numbers of positive and negative real zeros:
    f(x)=6x5−2x4−8x2+3x−1
  3. Possible numbers of positive and negative real zeros:
    f(x)=7x5+9x3+2x2+9x−2
  4. Possible numbers of positive and negative real zeros:
    f(x)=2x5+5x4−9x3−8x+4
  5. Possible numbers of positive and negative real zeros:
    f(x)=−6x5−9x4−9x2−7x+7
  6. Possible numbers of positive and negative real zeros:
    f(x)=−5x5+7x4+7x2−4x−9

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

Sigma Prep · sigmaprep.io/worksheets

Name

Descartes' Rule of SignsAnswer keyVersion 2

Date Period

Answer each question about the zeros of the polynomial.

  1. Find the possible positive real zeros

    Possible numbers of positive real zeros:
    f(x)=2x5+2x4−4x3−5x+3    2 or 0
  2. Possible numbers of positive real zeros:
    f(x)=−9x5+8x4−8x3−7x2+1    3 or 1
  3. Possible numbers of positive real zeros:
    f(x)=−4x5−7x4−8x3−4x+2    1
  4. Possible numbers of positive real zeros:
    f(x)=−2x5+5x3−3x2−8x−5    2 or 0
  5. Possible numbers of positive real zeros:
    f(x)=7x5−x4−3x2−9x−2    1
  6. Possible numbers of positive real zeros:
    f(x)=−8x5−x4+2x2−5x+1    3 or 1
  7. Possible numbers of positive real zeros:
    f(x)=−2x5−9x4−x3−4x2+1    1

  1. Find the possible negative real zeros

    Possible numbers of negative real zeros:
    f(x)=−x5−9x3−9x2−7x+6    2 or 0
  2. Possible numbers of negative real zeros:
    f(x)=9x5+3x4+8x3+6x+9    3 or 1
  3. Possible numbers of negative real zeros:
    f(x)=−3x5+6x4−7x2+3x+5    2 or 0
  4. Possible numbers of negative real zeros:
    f(x)=−3x5−8x4−8x2−2x+5    2 or 0
  5. Possible numbers of negative real zeros:
    f(x)=−4x5+2x3+7x2+x+4    4, 2 or 0
  6. Possible numbers of negative real zeros:
    f(x)=3x5−7x4−6x3−5x2+6    3 or 1
  7. Possible numbers of negative real zeros:
    f(x)=6x5+x4+8x3+3x2−6    4, 2 or 0

  1. Find possible positive and negative real zeros

    Possible numbers of positive and negative real zeros:
    f(x)=8x5−x4−5x3−4x2−6    Positive: 1; negative: 2 or 0
  2. Possible numbers of positive and negative real zeros:
    f(x)=6x5−2x4−8x2+3x−1    Positive: 3 or 1; negative: 0
  3. Possible numbers of positive and negative real zeros:
    f(x)=7x5+9x3+2x2+9x−2    Positive: 1; negative: 2 or 0
  4. Possible numbers of positive and negative real zeros:
    f(x)=2x5+5x4−9x3−8x+4    Positive: 2 or 0; negative: 1
  5. Possible numbers of positive and negative real zeros:
    f(x)=−6x5−9x4−9x2−7x+7    Positive: 1; negative: 2 or 0
  6. Possible numbers of positive and negative real zeros:
    f(x)=−5x5+7x4+7x2−4x−9    Positive: 2 or 0; negative: 1

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

How to do these

Possible numbers of positive and negative real zeros:
f(x)=x4−3x3+2x2−x−5

  1. Signs of f(x): + - + - -, three changes, so 3 or 1 positive zeros.
  2. f(-x) = x⁴ + 3x³ + 2x² + x - 5 has one change, so 1 negative zero.

The answer is Positive: 3 or 1; negative: 1.

Where students go wrong

Flipping the sign of every term to get f(-x). Only the odd powers change sign; x⁴, 2x² and the constant keep their signs.

Also called descartes rule of signs, fundamental theorem of algebra, possible positive and negative zeros or number of imaginary zeros.

What you can put on this worksheet

Possible positive real zeros
Possible numbers of positive real zeros: f(x)=−9x3−6x2+x+6 → 1
Possible negative real zeros
Possible numbers of negative real zeros: f(x)=−9x3−6x2+x+6 → 2 or 0
Both at once
Possible numbers of positive and negative real zeros: f(x)=−9x3−6x2+x+6 → Positive: 1; negative: 2 or 0
Every combination, with imaginary zeros
List every possible (positive, negative, imaginary) count of zeros: f(x)=−9x3−6x2+x+6 → (1,2,0),  (1,0,2)
Even powers, two terms, no constant, degree 6-9, and backwards
State the possible numbers of positive and negative real zeros: −8m3+3m2 → Positive: 1Negative: 0
How many complex zeros
How many complex zeros does f(x)=−4x5+9x4−8x2−9x have, counting multiplicity? → 5

Questions about these worksheets

Yes. Every descartes' rule of signs 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 descartes' rule of signs 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: possible positive real zeros, possible negative real zeros, both at once, every combination, with imaginary zeros, even powers, two terms, no constant, degree 6-9, and backwards and how many complex zeros. Tick as many as you want and set how many of each, or let it spread them evenly.

Counting a missing term as a sign change. Only the terms that are there count; skip straight over a zero coefficient to the next sign.