Worksheets · Precalculus

Zeros and multiplicity worksheet for Precalculus

Each factor gives a zero, and the power on that factor says how many times it counts. That number is what decides whether the graph crosses at the zero or turns round and comes back.

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.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Zeros and Multiplicity

Date Period

Find the zeros of each function and their multiplicities, or answer as asked.

  1. Find the zeros and their multiplicities

    f(x)=(x−5)(x−1)2
  2. f(x)=(x+2)2(x+4)
  3. f(x)=(x−5)(x−1)
  4. f(x)=(x−1)(x+1)2
  5. f(x)=(x+4)2(x+5)2
  6. f(x)=(x−2)2(x+5)
  7. f(x)=(x−3)2(x+1)2

  1. Find the zeros: crosses or touches

    f(x)=(x+6)2(x+1). Find the zeros and say whether the graph crosses or touches at each.
  2. f(x)=(x−6)(x+5)2. Find the zeros and say whether the graph crosses or touches at each.
  3. f(x)=(x+5)2(x−2). Find the zeros and say whether the graph crosses or touches at each.
  4. f(x)=x2(x−3). Find the zeros and say whether the graph crosses or touches at each.
  5. f(x)=(x−4)2(x+5). Find the zeros and say whether the graph crosses or touches at each.
  6. f(x)=(x−1)x2. Find the zeros and say whether the graph crosses or touches at each.
  7. f(x)=(x+1)(x−3)2. Find the zeros and say whether the graph crosses or touches at each.

  1. Factor to find zeros and multiplicities

    Find the zeros and their multiplicities:
    f(x)=2x4−6x3+6x2−2x
  2. Find the zeros and their multiplicities: f(x)=−x2+2x−1
  3. Find the zeros and their multiplicities: f(x)=x3−6x2+12x−8
  4. Find the zeros and their multiplicities: f(x)=x2−6x+9
  5. Find the zeros and their multiplicities: f(x)=x2+3x
  6. Find the zeros and their multiplicities: f(x)=−x4−8x3−16x2

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

Sigma Prep · sigmaprep.io/worksheets

Name

Zeros and MultiplicityAnswer keyVersion 1

Date Period

Find the zeros of each function and their multiplicities, or answer as asked.

  1. Find the zeros and their multiplicities

    f(x)=(x−5)(x−1)21 (multiplicity 2), 5 (multiplicity 1)
  2. f(x)=(x+2)2(x+4)−4 (multiplicity 1), −2 (multiplicity 2)
  3. f(x)=(x−5)(x−1)1 (multiplicity 1), 5 (multiplicity 1)
  4. f(x)=(x−1)(x+1)2−1 (multiplicity 2), 1 (multiplicity 1)
  5. f(x)=(x+4)2(x+5)2−5 (multiplicity 2), −4 (multiplicity 2)
  6. f(x)=(x−2)2(x+5)−5 (multiplicity 1), 2 (multiplicity 2)
  7. f(x)=(x−3)2(x+1)2−1 (multiplicity 2), 3 (multiplicity 2)

  1. Find the zeros: crosses or touches

    f(x)=(x+6)2(x+1). Find the zeros and say whether the graph crosses or touches at each.−6 (touches), −1 (crosses)
  2. f(x)=(x−6)(x+5)2. Find the zeros and say whether the graph crosses or touches at each.−5 (touches), 6 (crosses)
  3. f(x)=(x+5)2(x−2). Find the zeros and say whether the graph crosses or touches at each.−5 (touches), 2 (crosses)
  4. f(x)=x2(x−3). Find the zeros and say whether the graph crosses or touches at each.0 (touches), 3 (crosses)
  5. f(x)=(x−4)2(x+5). Find the zeros and say whether the graph crosses or touches at each.−5 (crosses), 4 (touches)
  6. f(x)=(x−1)x2. Find the zeros and say whether the graph crosses or touches at each.0 (touches), 1 (crosses)
  7. f(x)=(x+1)(x−3)2. Find the zeros and say whether the graph crosses or touches at each.−1 (crosses), 3 (touches)

  1. Factor to find zeros and multiplicities

    Find the zeros and their multiplicities:
    f(x)=2x4−6x3+6x2−2x    0 (multiplicity 1), 1 (multiplicity 3)
  2. Find the zeros and their multiplicities: f(x)=−x2+2x−11 (multiplicity 2)
  3. Find the zeros and their multiplicities: f(x)=x3−6x2+12x−82 (multiplicity 3)
  4. Find the zeros and their multiplicities: f(x)=x2−6x+93 (multiplicity 2)
  5. Find the zeros and their multiplicities: f(x)=x2+3x−3 (multiplicity 1), 0 (multiplicity 1)
  6. Find the zeros and their multiplicities: f(x)=−x4−8x3−16x2−4 (multiplicity 2), 0 (multiplicity 2)

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

How to do these

f(x)=(x−2)2(x+3)(x−1)3

  1. Set each factor to zero.
  2. The powers are 2, 1 and 3.
  3. Read the zeros off with those powers.

The answer is −3 (multiplicity 1), 1 (multiplicity 3), 2 (multiplicity 2).

Where students go wrong

Reading the zero straight off the factor. (x - 2) is zero at positive 2, not negative 2. The sign in the parenthesis is the opposite of the zero it gives you.

Also called multiplicity of zeros, roots of polynomials or zeros of polynomial functions.

What you can put on this worksheet

Zeros and their multiplicity
f(x)=(x−3)(x−1)2 → 1 (multiplicity 2), 3 (multiplicity 1)
Crosses or touches at each zero
f(x)=x2(x−1). Find the zeros and say whether the graph crosses or touches at each. → 0 (touches), 1 (crosses)
From an unfactored polynomial
Find the zeros and their multiplicities: f(x)=x3−6x2+9x → 0 (multiplicity 1), 3 (multiplicity 2)
Zeros, a degree and a sign condition
Write a polynomial of degree 4, leading coefficient ±1, with zeros 0 and 1 and f(x)≤0 for every x → f(x)=−x2(x−1)2

Questions about these worksheets

Yes. Every zeros and multiplicity 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 zeros and multiplicity 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 two 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 4: zeros and their multiplicity, crosses or touches at each zero, from an unfactored polynomial and zeros, a degree and a sign condition. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the zero straight off the factor. (x - 2) is zero at positive 2, not negative 2. The sign in the parenthesis is the opposite of the zero it gives you.