Write the polynomial of least degree with leading coefficient 1 that has zeros 6, -1, 9, -8

x^4 - 6x^3 - 73x^2 + 366x + 432

Write the polynomial of least degree with leading coefficient 1 that has zeros 8, -1, -6, 3

x^4 - 4x^3 - 47x^2 + 102x + 144

Write the polynomial of least degree with leading coefficient 1 that has zeros -4, -3, 3, -7

x^4 + 11x^3 + 19x^2 - 99x - 252

Write a polynomial function of least degree with integer coefficients that has the given zeros: -7/4, 4, -5/3

f(x) = 12x^3 - 7x^2 - 129x - 140

sqrt(5) is a zero of f(x) = x^4 + 2x^3 - 13x^2 - 10x + 40. Factor f into linear factors.

(x - sqrt(5))(x + sqrt(5))(x + 4)(x - 2)

Worksheets · Precalculus

Factors and zeros worksheet for Precalculus

Precalculus adds zeros that arrive in pairs: an irrational zero like 2 - √2 or a complex one like 3 + 2i brings its conjugate along, so a polynomial of least degree can need more factors than the zeros you were given. With one factor or root known, divide it out and finish with the quadratic.

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.



Sigma Prep · sigmaprep.io/worksheets Name Factors and Zeros Date Period Answer each question. Write a polynomial from its zeros Write the polynomial of least degree with leading coefficient 1 that has zeros $- 3, - 4, 1, 7$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zero $2 + 2 i$ Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, 3$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 5$ and $3 + \sqrt{2}$ Write the polynomial of least degree with leading coefficient 1 that has zeros $6, - 1, 9, - 8$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $1 + 5 i$ Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, - 6, 3$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 4$ and $- 3 + \sqrt{2}$ Write the polynomial of least degree with leading coefficient 1 that has zeros $- 4, - 3, 3, - 7$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $2 + \sqrt{2}$ Use a given factor to solve Solve $a^{4} + 5 a^{3} - 29 a^{2} - 105 a = 0$ , given that $a$ is a factor Solve $y^{4} + 17 y^{3} + 95 y^{2} + 199 y + 120 = 0$ , given that $y + 5$ is a factor Solve $r^{4} - 18 r^{3} + 99 r^{2} - 162 r = 0$ , given that $r - 9$ is a factor Solve $a^{4} - 2 a^{3} - 64 a^{2} + 32 a + 768 = 0$ , given that $a - 4$ is a factor Solve $n^{4} - 8 n^{3} - 43 n^{2} + 362 n - 504 = 0$ , given that $n - 4$ is a factor Factor given a complex zero Factor completely over the real numbers, given that $x = 1 + i$ is a zero: $x^{3} + 10 x^{2} - 22 x + 24$ Factor completely over the real numbers, given that $x = 5 - 3 i$ is a zero: $x^{3} + 7 x^{2} - 136 x + 578$ Factor completely over the real numbers, given that $x = - 1 + 2 i$ is a zero: $x^{3} + 19 x^{2} + 39 x + 85$ Factor completely over the real numbers, given that $x = - 4 + 2 i$ is a zero: $x^{3} - 5 x^{2} - 84 x - 260$ Factor completely over the real numbers, given that $x = 1 + 2 i$ is a zero: $x^{3} + 13 x^{2} - 25 x + 75$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Factors and Zeros

Date Period

Answer each question.

Write a polynomial from its zeros
Write the polynomial of least degree with leading coefficient 1 that has zeros $- 3, - 4, 1, 7$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zero $2 + 2 i$
Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, 3$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 5$ and $3 + \sqrt{2}$
Write the polynomial of least degree with leading coefficient 1 that has zeros $6, - 1, 9, - 8$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $1 + 5 i$
Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, - 6, 3$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 4$ and $- 3 + \sqrt{2}$
Write the polynomial of least degree with leading coefficient 1 that has zeros $- 4, - 3, 3, - 7$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $2 + \sqrt{2}$

Use a given factor to solve
Solve $a^{4} + 5 a^{3} - 29 a^{2} - 105 a = 0$ , given that $a$ is a factor
Solve $y^{4} + 17 y^{3} + 95 y^{2} + 199 y + 120 = 0$ , given that $y + 5$ is a factor
Solve $r^{4} - 18 r^{3} + 99 r^{2} - 162 r = 0$ , given that $r - 9$ is a factor
Solve $a^{4} - 2 a^{3} - 64 a^{2} + 32 a + 768 = 0$ , given that $a - 4$ is a factor
Solve $n^{4} - 8 n^{3} - 43 n^{2} + 362 n - 504 = 0$ , given that $n - 4$ is a factor

Factor given a complex zero
Factor completely over the real numbers, given that $x = 1 + i$ is a zero: $x^{3} + 10 x^{2} - 22 x + 24$
Factor completely over the real numbers, given that $x = 5 - 3 i$ is a zero: $x^{3} + 7 x^{2} - 136 x + 578$
Factor completely over the real numbers, given that $x = - 1 + 2 i$ is a zero: $x^{3} + 19 x^{2} + 39 x + 85$
Factor completely over the real numbers, given that $x = - 4 + 2 i$ is a zero: $x^{3} - 5 x^{2} - 84 x - 260$
Factor completely over the real numbers, given that $x = 1 + 2 i$ is a zero: $x^{3} + 13 x^{2} - 25 x + 75$



Sigma Prep · sigmaprep.io/worksheets Name Factors and ZerosAnswer keyVersion 2 Date Period Answer each question. Write a polynomial from its zeros Write the polynomial of least degree with leading coefficient 1 that has zeros $- 3, - 4, 1, 7$ $x^{4} - x^{3} - 37 x^{2} - 47 x + 84$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zero $2 + 2 i$ $x^{2} - 4 x + 8$ Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, 3$ $x^{3} - 10 x^{2} + 13 x + 24$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 5$ and $3 + \sqrt{2}$ $x^{3} - x^{2} - 23 x + 35$ Write the polynomial of least degree with leading coefficient 1 that has zeros $6, - 1, 9, - 8$ $x^{4} - 6 x^{3} - 73 x^{2} + 366 x + 432$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $1 + 5 i$ $x^{3} - 7 x^{2} + 36 x - 130$ Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, - 6, 3$ $x^{4} - 4 x^{3} - 47 x^{2} + 102 x + 144$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 4$ and $- 3 + \sqrt{2}$ $x^{3} + 10 x^{2} + 31 x + 28$ Write the polynomial of least degree with leading coefficient 1 that has zeros $- 4, - 3, 3, - 7$ $x^{4} + 11 x^{3} + 19 x^{2} - 99 x - 252$ Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $2 + \sqrt{2}$ $x^{3} - 9 x^{2} + 22 x - 10$ Use a given factor to solve Solve $a^{4} + 5 a^{3} - 29 a^{2} - 105 a = 0$ , given that $a$ is a factor ${- 7, - 3, 0, 5}$ Solve $y^{4} + 17 y^{3} + 95 y^{2} + 199 y + 120 = 0$ , given that $y + 5$ is a factor ${- 8, - 5, - 3, - 1}$ Solve $r^{4} - 18 r^{3} + 99 r^{2} - 162 r = 0$ , given that $r - 9$ is a factor ${0, 3, 6, 9}$ Solve $a^{4} - 2 a^{3} - 64 a^{2} + 32 a + 768 = 0$ , given that $a - 4$ is a factor ${- 6, - 4, 4, 8}$ Solve $n^{4} - 8 n^{3} - 43 n^{2} + 362 n - 504 = 0$ , given that $n - 4$ is a factor ${- 7, 2, 4, 9}$ Factor given a complex zero Factor completely over the real numbers, given that $x = 1 + i$ is a zero: $x^{3} + 10 x^{2} - 22 x + 24$ $(x + 12) (x^{2} - 2 x + 2)$ Factor completely over the real numbers, given that $x = 5 - 3 i$ is a zero: $x^{3} + 7 x^{2} - 136 x + 578$ $(x + 17) (x^{2} - 10 x + 34)$ Factor completely over the real numbers, given that $x = - 1 + 2 i$ is a zero: $x^{3} + 19 x^{2} + 39 x + 85$ $(x + 17) (x^{2} + 2 x + 5)$ Factor completely over the real numbers, given that $x = - 4 + 2 i$ is a zero: $x^{3} - 5 x^{2} - 84 x - 260$ $(x - 13) (x^{2} + 8 x + 20)$ Factor completely over the real numbers, given that $x = 1 + 2 i$ is a zero: $x^{3} + 13 x^{2} - 25 x + 75$ $(x + 15) (x^{2} - 2 x + 5)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Factors and ZerosAnswer keyVersion 2

Date Period

Answer each question.

Write a polynomial from its zeros
Write the polynomial of least degree with leading coefficient 1 that has zeros $- 3, - 4, 1, 7$ $x^{4} - x^{3} - 37 x^{2} - 47 x + 84$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zero $2 + 2 i$ $x^{2} - 4 x + 8$
Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, 3$ $x^{3} - 10 x^{2} + 13 x + 24$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 5$ and $3 + \sqrt{2}$ $x^{3} - x^{2} - 23 x + 35$
Write the polynomial of least degree with leading coefficient 1 that has zeros $6, - 1, 9, - 8$ $x^{4} - 6 x^{3} - 73 x^{2} + 366 x + 432$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $1 + 5 i$ $x^{3} - 7 x^{2} + 36 x - 130$
Write the polynomial of least degree with leading coefficient 1 that has zeros $8, - 1, - 6, 3$ $x^{4} - 4 x^{3} - 47 x^{2} + 102 x + 144$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $- 4$ and $- 3 + \sqrt{2}$ $x^{3} + 10 x^{2} + 31 x + 28$
Write the polynomial of least degree with leading coefficient 1 that has zeros $- 4, - 3, 3, - 7$ $x^{4} + 11 x^{3} + 19 x^{2} - 99 x - 252$
Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $5$ and $2 + \sqrt{2}$ $x^{3} - 9 x^{2} + 22 x - 10$

Use a given factor to solve
Solve $a^{4} + 5 a^{3} - 29 a^{2} - 105 a = 0$ , given that $a$ is a factor ${- 7, - 3, 0, 5}$
Solve $y^{4} + 17 y^{3} + 95 y^{2} + 199 y + 120 = 0$ , given that $y + 5$ is a factor ${- 8, - 5, - 3, - 1}$
Solve $r^{4} - 18 r^{3} + 99 r^{2} - 162 r = 0$ , given that $r - 9$ is a factor ${0, 3, 6, 9}$
Solve $a^{4} - 2 a^{3} - 64 a^{2} + 32 a + 768 = 0$ , given that $a - 4$ is a factor ${- 6, - 4, 4, 8}$
Solve $n^{4} - 8 n^{3} - 43 n^{2} + 362 n - 504 = 0$ , given that $n - 4$ is a factor ${- 7, 2, 4, 9}$

Factor given a complex zero
Factor completely over the real numbers, given that $x = 1 + i$ is a zero: $x^{3} + 10 x^{2} - 22 x + 24$ $(x + 12) (x^{2} - 2 x + 2)$
Factor completely over the real numbers, given that $x = 5 - 3 i$ is a zero: $x^{3} + 7 x^{2} - 136 x + 578$ $(x + 17) (x^{2} - 10 x + 34)$
Factor completely over the real numbers, given that $x = - 1 + 2 i$ is a zero: $x^{3} + 19 x^{2} + 39 x + 85$ $(x + 17) (x^{2} + 2 x + 5)$
Factor completely over the real numbers, given that $x = - 4 + 2 i$ is a zero: $x^{3} - 5 x^{2} - 84 x - 260$ $(x - 13) (x^{2} + 8 x + 20)$
Factor completely over the real numbers, given that $x = 1 + 2 i$ is a zero: $x^{3} + 13 x^{2} - 25 x + 75$ $(x + 15) (x^{2} - 2 x + 5)$

How to do these

Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zeros $2$ and $1 + i$

Complex zeros come in conjugate pairs, so 1 - i is a zero too.
(x - (1 + i))(x - (1 - i)) = x² - 2x + 2.
Multiply by (x - 2).

The answer is $x^{3} - 4 x^{2} + 6 x - 4$ .

Where students go wrong

Leaving out the conjugate. With integer coefficients, 1 + i cannot be a zero unless 1 - i is as well, so the least degree here is 3, and a quadratic built from 2 and 1 + i alone has an i in its coefficients.

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 the polynomial of least degree with leading coefficient 1 that has zeros $1, 3$ → $x^{2} - 4 x + 3$
One factor given, find the other zeros: Solve $v^{3} + v^{2} - 17 v + 15 = 0$ , given that $v - 1$ is a factor → ${- 5, 1, 3}$
A polynomial from its roots: Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has roots $- 3$ and $2 + i$ . → $f (x) = x^{3} - x^{2} - 7 x + 15$
From irrational or complex zeros: Write the polynomial of least degree with integer coefficients and leading coefficient 1 that has zero $\sqrt{7}$ → $x^{2} - 7$
Factor, given one complex root: Factor completely over the real numbers, given that $x = 2 + i$ is a zero: $x^{3} - 2 x^{2} - 3 x + 10$ → $(x + 2) (x^{2} - 4 x + 5)$
Fraction and complex zeros, conjugates, multiplicity, a given factor: Write a polynomial function of least degree with integer coefficients that has the given zeros:
$- \frac{7}{4}, 4, - \frac{5}{3}$ → $f (x) = 12 x^{3} - 7 x^{2} - 129 x - 140$
A zero of multiplicity 3: Write a polynomial of least degree with leading coefficient 1 and these zeros:
$0 (multiplicity 3), 1$ → $f (x) = x^{3} (x - 1) = x^{4} - x^{3}$
Given one irrational zero: $\sqrt{5}$ is a zero of $f (x) = x^{4} + 2 x^{3} - 13 x^{2} - 10 x + 40$ . Factor $f$ into linear factors. → $(x - \sqrt{5}) (x + \sqrt{5}) (x + 4) (x - 2)$

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.

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 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 8: write a polynomial from its zeros, one factor given, find the other zeros, a polynomial from its roots, from irrational or complex zeros, factor, given one complex root, fraction and complex zeros, conjugates, multiplicity, a given factor, a zero of multiplicity 3 and given one irrational zero. 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.

Factors and zeros worksheet for Precalculus

Select your difficulty

Write a polynomial from its zeros

Use a given factor to solve

Factor given a complex zero

Write a polynomial from its zeros

Use a given factor to solve

Factor given a complex zero

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these factors and zeros worksheets free?+

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What grade are these factors and zeros worksheets for?+

Can I get a different version of the same factors and zeros worksheet?+

Is this a free alternative to Kuta Software for factors and zeros?+

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