List the possible rational zeros of f(x) = 3x^3 - 4x^2 - 6x - 10

plus or minus 1/3, plus or minus 2/3, plus or minus 1, plus or minus 5/3, plus or minus 2, plus or minus 10/3, plus or minus 5, plus or minus 10

List the possible rational zeros of f(x) = 6x^3 - x - 7

plus or minus 1/6, plus or minus 1/3, plus or minus 1/2, plus or minus 1, plus or minus 7/6, plus or minus 7/3, plus or minus 7/2, plus or minus 7

List the possible rational zeros of f(x) = 6x^3 + 6x^2 - 5x - 12

plus or minus 1/6, plus or minus 1/3, plus or minus 1/2, plus or minus 2/3, plus or minus 1, plus or minus 4/3, plus or minus 3/2, plus or minus 2, plus or minus 3, plus or minus 4, plus or minus 6, plus or minus 12

List the possible rational zeros of f(x) = 4x^3 + 4x^2 + 2x - 5

plus or minus 1/4, plus or minus 1/2, plus or minus 1, plus or minus 5/4, plus or minus 5/2, plus or minus 5

List the possible rational zeros of f(x) = 4x^3 + 3x^2 + 4x + 2

plus or minus 1/4, plus or minus 1/2, plus or minus 1, plus or minus 2

List the possible rational zeros of f(x) = 2x^3 - 2x^2 + 6x + 9

plus or minus 1/2, plus or minus 1, plus or minus 3/2, plus or minus 3, plus or minus 9/2, plus or minus 9

List the possible rational zeros of f(x) = 4x^3 - 3x + 7

plus or minus 1/4, plus or minus 1/2, plus or minus 1, plus or minus 7/4, plus or minus 7/2, plus or minus 7

Find all real and complex zeros of f(x) = 2x^3 + 7x^2 - 18x + 7

1/2, -2 + sqrt(11), -2 - sqrt(11)

Find all real and complex zeros of f(x) = 3x^3 - 8x^2 + 14x + 40

-4/3, 2 + isqrt(6), 2 - isqrt(6)

Find all real and complex zeros of f(x) = 3x^3 - 8x^2 + 16x - 8

2/3, 1 + isqrt(3), 1 - isqrt(3)

Find all real and complex zeros of f(x) = 3x^3 - 2x^2 - 29x - 28

-4/3, 1 + 2sqrt(2), 1 - 2sqrt(2)

Find all real and complex zeros of f(x) = 3x^3 - 13x^2 - 8x + 4

1/3, 2 + 2sqrt(2), 2 - 2sqrt(2)

List the possible rational zeros of f(x) = x^2 - 4x + 10

plus or minus 1, plus or minus 2, plus or minus 5, plus or minus 10

Worksheets · Precalculus

Rational root theorem worksheet for Precalculus

Precalculus pushes the rational root theorem to cubics with a leading coefficient other than 1, where not every zero is rational. List the candidates, test them with synthetic division, and once one works, solve the quadratic that is left, which may give irrational or complex zeros.

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 Rational Zeros Date Period Solve each equation, or answer as asked. Solve using the rational root theorem $3 r^{3} - 19 r^{2} - 4 r + 60 = 0$ $2 n^{3} + 7 n^{2} - 13 n - 18 = 0$ $2 v^{3} - 25 v^{2} + 82 v - 35 = 0$ $5 a^{3} + 33 a^{2} + 46 a - 24 = 0$ $2 n^{3} + n^{2} - 73 n + 126 = 0$ $3 m^{3} + 41 m^{2} + 178 m + 240 = 0$ $3 m^{3} + 11 m^{2} - 67 m + 21 = 0$ List the possible rational zeros List the possible rational zeros of $f (x) = 3 x^{3} - 4 x^{2} - 6 x - 10$ List the possible rational zeros of $f (x) = 6 x^{3} - x - 7$ List the possible rational zeros of $f (x) = 6 x^{3} + 6 x^{2} - 5 x - 12$ List the possible rational zeros of $f (x) = 4 x^{3} + 4 x^{2} + 2 x - 5$ List the possible rational zeros of $f (x) = 4 x^{3} + 3 x^{2} + 4 x + 2$ List the possible rational zeros of $f (x) = 2 x^{3} - 2 x^{2} + 6 x + 9$ List the possible rational zeros of $f (x) = 4 x^{3} - 3 x + 7$ Find all zeros Find all real and complex zeros of $f (x) = 2 x^{3} + 7 x^{2} - 18 x + 7$ Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 14 x + 40$ Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 16 x - 8$ Find all real and complex zeros of $f (x) = 3 x^{3} - 2 x^{2} - 29 x - 28$ Find all real and complex zeros of $f (x) = 3 x^{3} - 13 x^{2} - 8 x + 4$ Find all real and complex zeros of $f (x) = 3 x^{3} - x^{2} - 15 x + 5$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational Zeros

Date Period

Solve each equation, or answer as asked.

Solve using the rational root theorem
$3 r^{3} - 19 r^{2} - 4 r + 60 = 0$
$2 n^{3} + 7 n^{2} - 13 n - 18 = 0$
$2 v^{3} - 25 v^{2} + 82 v - 35 = 0$
$5 a^{3} + 33 a^{2} + 46 a - 24 = 0$
$2 n^{3} + n^{2} - 73 n + 126 = 0$
$3 m^{3} + 41 m^{2} + 178 m + 240 = 0$
$3 m^{3} + 11 m^{2} - 67 m + 21 = 0$

List the possible rational zeros
List the possible rational zeros of $f (x) = 3 x^{3} - 4 x^{2} - 6 x - 10$
List the possible rational zeros of $f (x) = 6 x^{3} - x - 7$
List the possible rational zeros of $f (x) = 6 x^{3} + 6 x^{2} - 5 x - 12$
List the possible rational zeros of $f (x) = 4 x^{3} + 4 x^{2} + 2 x - 5$
List the possible rational zeros of $f (x) = 4 x^{3} + 3 x^{2} + 4 x + 2$
List the possible rational zeros of $f (x) = 2 x^{3} - 2 x^{2} + 6 x + 9$
List the possible rational zeros of $f (x) = 4 x^{3} - 3 x + 7$

Find all zeros
Find all real and complex zeros of $f (x) = 2 x^{3} + 7 x^{2} - 18 x + 7$
Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 14 x + 40$
Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 16 x - 8$
Find all real and complex zeros of $f (x) = 3 x^{3} - 2 x^{2} - 29 x - 28$
Find all real and complex zeros of $f (x) = 3 x^{3} - 13 x^{2} - 8 x + 4$
Find all real and complex zeros of $f (x) = 3 x^{3} - x^{2} - 15 x + 5$



Sigma Prep · sigmaprep.io/worksheets Name Rational ZerosAnswer keyVersion 2 Date Period Solve each equation, or answer as asked. Solve using the rational root theorem $3 r^{3} - 19 r^{2} - 4 r + 60 = 0$ ${- \frac{5}{3}, 2, 6}$ $2 n^{3} + 7 n^{2} - 13 n - 18 = 0$ ${- \frac{9}{2}, - 1, 2}$ $2 v^{3} - 25 v^{2} + 82 v - 35 = 0$ ${\frac{1}{2}, 5, 7}$ $5 a^{3} + 33 a^{2} + 46 a - 24 = 0$ ${- 4, - 3, \frac{2}{5}}$ $2 n^{3} + n^{2} - 73 n + 126 = 0$ ${- 7, 2, \frac{9}{2}}$ $3 m^{3} + 41 m^{2} + 178 m + 240 = 0$ ${- 6, - 5, - \frac{8}{3}}$ $3 m^{3} + 11 m^{2} - 67 m + 21 = 0$ ${- 7, \frac{1}{3}, 3}$ List the possible rational zeros List the possible rational zeros of $f (x) = 3 x^{3} - 4 x^{2} - 6 x - 10$ $\pm \frac{1}{3},$ $\pm \frac{2}{3},$ $\pm 1,$ $\pm \frac{5}{3},$ $\pm 2,$ $\pm \frac{10}{3},$ $\pm 5,$ $\pm 10$ List the possible rational zeros of $f (x) = 6 x^{3} - x - 7$ $\pm \frac{1}{6},$ $\pm \frac{1}{3},$ $\pm \frac{1}{2},$ $\pm 1,$ $\pm \frac{7}{6},$ $\pm \frac{7}{3},$ $\pm \frac{7}{2},$ $\pm 7$ List the possible rational zeros of $f (x) = 6 x^{3} + 6 x^{2} - 5 x - 12$ $\pm \frac{1}{6},$ $\pm \frac{1}{3},$ $\pm \frac{1}{2},$ $\pm \frac{2}{3},$ $\pm 1,$ $\pm \frac{4}{3},$ $\pm \frac{3}{2},$ $\pm 2,$ $\pm 3,$ $\pm 4,$ $\pm 6,$ $\pm 12$ List the possible rational zeros of $f (x) = 4 x^{3} + 4 x^{2} + 2 x - 5$ $\pm \frac{1}{4}, \pm \frac{1}{2}, \pm 1, \pm \frac{5}{4}, \pm \frac{5}{2}, \pm 5$ List the possible rational zeros of $f (x) = 4 x^{3} + 3 x^{2} + 4 x + 2$ $\pm \frac{1}{4}, \pm \frac{1}{2}, \pm 1, \pm 2$ List the possible rational zeros of $f (x) = 2 x^{3} - 2 x^{2} + 6 x + 9$ $\pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 3, \pm \frac{9}{2}, \pm 9$ List the possible rational zeros of $f (x) = 4 x^{3} - 3 x + 7$ $\pm \frac{1}{4}, \pm \frac{1}{2}, \pm 1, \pm \frac{7}{4}, \pm \frac{7}{2}, \pm 7$ Find all zeros Find all real and complex zeros of $f (x) = 2 x^{3} + 7 x^{2} - 18 x + 7$ ${\frac{1}{2},$ $- 2 + \sqrt{11},$ $- 2 - \sqrt{11}}$ Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 14 x + 40$ ${- \frac{4}{3}, 2 + i \sqrt{6}, 2 - i \sqrt{6}}$ Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 16 x - 8$ ${\frac{2}{3}, 1 + i \sqrt{3}, 1 - i \sqrt{3}}$ Find all real and complex zeros of $f (x) = 3 x^{3} - 2 x^{2} - 29 x - 28$ ${- \frac{4}{3}, 1 + 2 \sqrt{2}, 1 - 2 \sqrt{2}}$ Find all real and complex zeros of $f (x) = 3 x^{3} - 13 x^{2} - 8 x + 4$ ${\frac{1}{3}, 2 + 2 \sqrt{2}, 2 - 2 \sqrt{2}}$ Find all real and complex zeros of $f (x) = 3 x^{3} - x^{2} - 15 x + 5$ ${\frac{1}{3}, \sqrt{5}, - \sqrt{5}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational ZerosAnswer keyVersion 2

Date Period

Solve each equation, or answer as asked.

Solve using the rational root theorem
$3 r^{3} - 19 r^{2} - 4 r + 60 = 0$ ${- \frac{5}{3}, 2, 6}$
$2 n^{3} + 7 n^{2} - 13 n - 18 = 0$ ${- \frac{9}{2}, - 1, 2}$
$2 v^{3} - 25 v^{2} + 82 v - 35 = 0$ ${\frac{1}{2}, 5, 7}$
$5 a^{3} + 33 a^{2} + 46 a - 24 = 0$ ${- 4, - 3, \frac{2}{5}}$
$2 n^{3} + n^{2} - 73 n + 126 = 0$ ${- 7, 2, \frac{9}{2}}$
$3 m^{3} + 41 m^{2} + 178 m + 240 = 0$ ${- 6, - 5, - \frac{8}{3}}$
$3 m^{3} + 11 m^{2} - 67 m + 21 = 0$ ${- 7, \frac{1}{3}, 3}$

List the possible rational zeros
List the possible rational zeros of $f (x) = 3 x^{3} - 4 x^{2} - 6 x - 10$ $\pm \frac{1}{3},$ $\pm \frac{2}{3},$ $\pm 1,$ $\pm \frac{5}{3},$ $\pm 2,$ $\pm \frac{10}{3},$ $\pm 5,$ $\pm 10$
List the possible rational zeros of $f (x) = 6 x^{3} - x - 7$ $\pm \frac{1}{6},$ $\pm \frac{1}{3},$ $\pm \frac{1}{2},$ $\pm 1,$ $\pm \frac{7}{6},$ $\pm \frac{7}{3},$ $\pm \frac{7}{2},$ $\pm 7$
List the possible rational zeros of $f (x) = 6 x^{3} + 6 x^{2} - 5 x - 12$ $\pm \frac{1}{6},$ $\pm \frac{1}{3},$ $\pm \frac{1}{2},$ $\pm \frac{2}{3},$ $\pm 1,$ $\pm \frac{4}{3},$ $\pm \frac{3}{2},$ $\pm 2,$ $\pm 3,$ $\pm 4,$ $\pm 6,$ $\pm 12$
List the possible rational zeros of $f (x) = 4 x^{3} + 4 x^{2} + 2 x - 5$ $\pm \frac{1}{4}, \pm \frac{1}{2}, \pm 1, \pm \frac{5}{4}, \pm \frac{5}{2}, \pm 5$
List the possible rational zeros of $f (x) = 4 x^{3} + 3 x^{2} + 4 x + 2$ $\pm \frac{1}{4}, \pm \frac{1}{2}, \pm 1, \pm 2$
List the possible rational zeros of $f (x) = 2 x^{3} - 2 x^{2} + 6 x + 9$ $\pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 3, \pm \frac{9}{2}, \pm 9$
List the possible rational zeros of $f (x) = 4 x^{3} - 3 x + 7$ $\pm \frac{1}{4}, \pm \frac{1}{2}, \pm 1, \pm \frac{7}{4}, \pm \frac{7}{2}, \pm 7$

Find all zeros
Find all real and complex zeros of $f (x) = 2 x^{3} + 7 x^{2} - 18 x + 7$ ${\frac{1}{2},$ $- 2 + \sqrt{11},$ $- 2 - \sqrt{11}}$
Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 14 x + 40$ ${- \frac{4}{3}, 2 + i \sqrt{6}, 2 - i \sqrt{6}}$
Find all real and complex zeros of $f (x) = 3 x^{3} - 8 x^{2} + 16 x - 8$ ${\frac{2}{3}, 1 + i \sqrt{3}, 1 - i \sqrt{3}}$
Find all real and complex zeros of $f (x) = 3 x^{3} - 2 x^{2} - 29 x - 28$ ${- \frac{4}{3}, 1 + 2 \sqrt{2}, 1 - 2 \sqrt{2}}$
Find all real and complex zeros of $f (x) = 3 x^{3} - 13 x^{2} - 8 x + 4$ ${\frac{1}{3}, 2 + 2 \sqrt{2}, 2 - 2 \sqrt{2}}$
Find all real and complex zeros of $f (x) = 3 x^{3} - x^{2} - 15 x + 5$ ${\frac{1}{3}, \sqrt{5}, - \sqrt{5}}$

How to do these

Find all zeros of $f (x) = 2 x^{3} - 5 x^{2} - 2 x + 2$

The candidates are ±1, ±2 and ±1/2, and f(1/2) = 0.
Synthetic division by 1/2 leaves 2x² - 4x - 4, which is 2(x² - 2x - 2).
The quadratic formula on x² - 2x - 2 = 0 gives 1 ± √3.

The answer is ${\frac{1}{2}, 1 + \sqrt{3}, 1 - \sqrt{3}}$ .

Where students go wrong

Writing the fractions upside down. The top comes from the constant and the bottom from the leading coefficient, so for 4x³ + 2x² + 3x - 7 the candidate is 7/4, not 4/7.

Also called rational zero theorem, finding rational zeros or possible rational roots.

What you can put on this worksheet

Zeros that are fractions: $3 v^{2} - v = 0$ → ${0, \frac{1}{3}}$
List the possible rational zeros: List the possible rational zeros of $f (x) = x^{2} - 4 x + 10$ → $\pm 1, \pm 2, \pm 5, \pm 10$
All zeros, some irrational or complex: Find all real and complex zeros of $f (x) = x^{3} - 3 x^{2} - 7 x + 21$ → ${3, \sqrt{7}, - \sqrt{7}}$
Higher degree lists, quartics, a repeated zero: Find all rational zeros: $3 p^{3} - 22 p^{2} + 43 p - 12$ → $\frac{1}{3}, 3, 4$

Questions about these worksheets

Yes. Every rational root theorem 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 rational root theorem 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 4: zeros that are fractions, list the possible rational zeros, all zeros, some irrational or complex and higher degree lists, quartics, a repeated zero. Tick as many as you want and set how many of each, or let it spread them evenly.

Only testing whole numbers. If the leading coefficient is not 1 then a fractional zero is likely, and a student who tries 1, 2 and 3 and gives up concludes wrongly that it does not factor.

Rational root theorem worksheet for Precalculus

Select your difficulty

Solve using the rational root theorem

List the possible rational zeros

Find all zeros

Solve using the rational root theorem

List the possible rational zeros

Find all zeros

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these rational root theorem worksheets free?+

Do the rational root theorem worksheets come with an answer key?+

Are these rational root theorem worksheets printable?+

What grade are these rational root theorem worksheets for?+

Can I get a different version of the same rational root theorem worksheet?+

Is this a free alternative to Kuta Software for rational root theorem?+

Can I choose how hard the rational root theorem questions are?+

What kinds of question are on a rational root theorem worksheet?+

What do students get wrong with rational root theorem?+