Worksheets · Precalculus

Conjugate roots worksheet for Precalculus

A polynomial with whole-number coefficients cannot have just one of a conjugate pair as a root. If 2 minus i is a root then 2 plus i is too, and that is what lets you build the whole polynomial from part of the list. The other half of the sheet gives real zeros with their multiplicities and one point on the graph: build the factors, then use the point to find the number in front.

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.

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

Polynomials and Conjugate Roots

Date Period

Write each polynomial.

  1. Write a polynomial from its zeros

    Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 2 and 1+i.
  2. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −3 and 1−i.
  3. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −4 and 2+i.
  4. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −3 and −3+i.
  5. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 4 and −3−i.
  6. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −3 and −1−i.
  7. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 2 and 2−i.
  8. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 1 and −3+i.
  9. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 1 and −2+i.
  10. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −4 and −3+i.

  1. Write a polynomial from its zeros and a point

    Write the polynomial of least degree with zeros 3 and −5 that passes through (0,15).
  2. Write the polynomial of least degree with zeros 2 (multiplicity 2) and −4 that passes through (0,32).
  3. Write the polynomial of least degree with zeros 3 and −2 that passes through (2,−8).
  4. Write the polynomial of least degree with zeros 4 (multiplicity 2) and 5 that passes through (0,−240).
  5. Write the polynomial of least degree with zeros 1 and 5 that passes through (0,15).
  6. Write the polynomial of least degree with zeros −3 and 1 that passes through (−1,−12).
  7. Write the polynomial of least degree with zeros −2 and 4 that passes through (−1,10).
  8. Write the polynomial of least degree with zeros 2 and 5 that passes through (0,−30).
  9. Write the polynomial of least degree with zeros −3 and 5 that passes through (1,−48).
  10. Write the polynomial of least degree with zeros −3 and −2 that passes through (1,36).

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

Sigma Prep · sigmaprep.io/worksheets

Name

Polynomials and Conjugate RootsAnswer keyVersion 1

Date Period

Write each polynomial.

  1. Write a polynomial from its zeros

    Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 2 and 1+i.f(x)=x3−4x2+6x−4
  2. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −3 and 1−i.f(x)=x3+x2−4x+6
  3. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −4 and 2+i.f(x)=x3−11x+20
  4. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −3 and −3+i.f(x)=x3+9x2+28x+30
  5. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 4 and −3−i.f(x)=x3+2x2−14x−40
  6. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −3 and −1−i.f(x)=x3+5x2+8x+6
  7. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 2 and 2−i.f(x)=x3−6x2+13x−10
  8. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 1 and −3+i.f(x)=x3+5x2+4x−10
  9. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots 1 and −2+i.f(x)=x3+3x2+x−5
  10. Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −4 and −3+i.f(x)=x3+10x2+34x+40

  1. Write a polynomial from its zeros and a point

    Write the polynomial of least degree with zeros 3 and −5 that passes through (0,15).f(x)=−(x−3)(x+5)
  2. Write the polynomial of least degree with zeros 2 (multiplicity 2) and −4 that passes through (0,32).f(x)=2(x−2)2(x+4)
  3. Write the polynomial of least degree with zeros 3 and −2 that passes through (2,−8).f(x)=2(x−3)(x+2)
  4. Write the polynomial of least degree with zeros 4 (multiplicity 2) and 5 that passes through (0,−240).f(x)=3(x−4)2(x−5)
  5. Write the polynomial of least degree with zeros 1 and 5 that passes through (0,15).f(x)=3(x−1)(x−5)
  6. Write the polynomial of least degree with zeros −3 and 1 that passes through (−1,−12).f(x)=3(x+3)(x−1)
  7. Write the polynomial of least degree with zeros −2 and 4 that passes through (−1,10).f(x)=−2(x+2)(x−4)
  8. Write the polynomial of least degree with zeros 2 and 5 that passes through (0,−30).f(x)=−3(x−2)(x−5)
  9. Write the polynomial of least degree with zeros −3 and 5 that passes through (1,−48).f(x)=3(x+3)(x−5)
  10. Write the polynomial of least degree with zeros −3 and −2 that passes through (1,36).f(x)=3(x+3)(x+2)

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

How to do these

Least degree with roots 3 and 2−i

  1. 2 plus i has to be a root as well.
  2. The pair multiplies to x squared minus 4x plus 5.
  3. Multiply that by x minus 3.

The answer is x3−7x2+17x−15.

Where students go wrong

Using only the root you were given. One complex root on its own gives a polynomial with complex coefficients, which is not what the question wants. Bring its partner along.

Also called complex conjugate root theorem, imaginary roots or irrational conjugates.

What you can put on this worksheet

A polynomial from its roots
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots −3 and 2+i. → f(x)=x3−x2−7x+15
Factor, given one complex root
Factor completely over the real numbers, given that x=2+i is a zero: x3−2x2−3x+10 → (x+2)(x2−4x+5)
Through a given point
Write the polynomial of least degree with zeros 3 and 1 that passes through (0,−3). → f(x)=−(x−3)(x−1)
Through points, no zeros given
Write the polynomial f of least degree through (−1,−6), (0,−5), (2,9) → f(x)=2x2+3x−5
One conjugate pair given, one not
Write a polynomial of least degree with integer coefficients and these zeros:
2i, −2i, 2+i → f(x)=x4−4x3+9x2−16x+20

Questions about these worksheets

Yes. Every conjugate roots 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 conjugate roots 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 5: a polynomial from its roots, factor, given one complex root, through a given point, through points, no zeros given and one conjugate pair given, one not. Tick as many as you want and set how many of each, or let it spread them evenly.

Using only the root you were given. One complex root on its own gives a polynomial with complex coefficients, which is not what the question wants. Bring its partner along.