Use synthetic division to decide whether -4 is an upper bound, a lower bound, or neither for the real zeros of f(x) = 2x^3 - 5x^2 + 9x - 3

Bottom row 2, -13, 61, -247; -4 is a lower bound

Remainder theorem worksheet for Precalculus

In Precalculus the remainder theorem stands in for long division on cubics and quartics: evaluate f(c), by direct substitution or synthetic division, and that number is the remainder. A remainder of 0 settles the factor question, which is the first step toward finding all the 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 The Remainder Theorem Date Period Find each remainder, or answer as asked. Find the remainder Find the remainder when $- 12 r^{4} - 12 r^{3} + 11 r^{2} + 12 r - 10$ is divided by $r + 2$ . Find the remainder when $10 x^{4} - 11 x^{3} - 12 x^{2} - 10 x + 10$ is divided by $x + 2$ . Find the remainder when $12 a^{4} - 12 a^{3} - 12 a^{2} + 12 a - 12$ is divided by $a - 2$ . Find the remainder when $- 11 m^{4} + 11 m^{3} + 12 m^{2} - 11 m - 12$ is divided by $m - 3$ . Find the remainder when $10 b^{4} + 11 b^{3} - 11 b^{2} + 11 b + 10$ is divided by $b + 3$ . Find the remainder when $11 p^{4} - 12 p^{3} + 10 p^{2} - 11 p - 11$ is divided by $p + 1$ . Find the remainder when $10 k^{4} + 11 k^{3} + 11 k^{2} - 12 k - 10$ is divided by $k - 3$ . Find the remainder when $- 11 y^{4} + 12 y^{3} - 11 y^{2} + 11 y - 11$ is divided by $y - 1$ . Find the remainder when $- 11 p^{4} - 11 p^{3} + 10 p^{2} - 12 p + 12$ is divided by $p - 1$ . Find the remainder when $10 x^{4} + 11 x^{3} - 12 x^{2} + 11 x - 10$ is divided by $x + 3$ . Decide whether it is a factor Is $x - 3$ a factor of $x^{3} - x^{2} - x - 15$ ? Is $x + 5$ a factor of $x^{3} + 8 x^{2} + 12 x - 15$ ? Is $x + 5$ a factor of $x^{3} + 7 x^{2} + 14 x + 25$ ? Is $x - 2$ a factor of $x^{3} - 5 x^{2} + x + 10$ ? Is $x - 3$ a factor of $x^{3} - 9 x - 8$ ? Is $x + 1$ a factor of $x^{3} + 3 x^{2} - 2$ ? Is $x - 1$ a factor of $x^{3} - 6 x^{2} + 4$ ? Is $x + 3$ a factor of $x^{3} + 2 x^{2} - 2 x + 7$ ? Is $x - 2$ a factor of $x^{3} + 2 x^{2} - 12 x + 14$ ? Is $x - 5$ a factor of $x^{3} - 9 x^{2} + 23 x - 19$ ? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Remainder Theorem

Date Period

Find each remainder, or answer as asked.

Find the remainder
Find the remainder when $- 12 r^{4} - 12 r^{3} + 11 r^{2} + 12 r - 10$ is divided by $r + 2$ .
Find the remainder when $10 x^{4} - 11 x^{3} - 12 x^{2} - 10 x + 10$ is divided by $x + 2$ .
Find the remainder when $12 a^{4} - 12 a^{3} - 12 a^{2} + 12 a - 12$ is divided by $a - 2$ .
Find the remainder when $- 11 m^{4} + 11 m^{3} + 12 m^{2} - 11 m - 12$ is divided by $m - 3$ .
Find the remainder when $10 b^{4} + 11 b^{3} - 11 b^{2} + 11 b + 10$ is divided by $b + 3$ .
Find the remainder when $11 p^{4} - 12 p^{3} + 10 p^{2} - 11 p - 11$ is divided by $p + 1$ .
Find the remainder when $10 k^{4} + 11 k^{3} + 11 k^{2} - 12 k - 10$ is divided by $k - 3$ .
Find the remainder when $- 11 y^{4} + 12 y^{3} - 11 y^{2} + 11 y - 11$ is divided by $y - 1$ .
Find the remainder when $- 11 p^{4} - 11 p^{3} + 10 p^{2} - 12 p + 12$ is divided by $p - 1$ .
Find the remainder when $10 x^{4} + 11 x^{3} - 12 x^{2} + 11 x - 10$ is divided by $x + 3$ .

Decide whether it is a factor
Is $x - 3$ a factor of $x^{3} - x^{2} - x - 15$ ?
Is $x + 5$ a factor of $x^{3} + 8 x^{2} + 12 x - 15$ ?
Is $x + 5$ a factor of $x^{3} + 7 x^{2} + 14 x + 25$ ?
Is $x - 2$ a factor of $x^{3} - 5 x^{2} + x + 10$ ?
Is $x - 3$ a factor of $x^{3} - 9 x - 8$ ?
Is $x + 1$ a factor of $x^{3} + 3 x^{2} - 2$ ?
Is $x - 1$ a factor of $x^{3} - 6 x^{2} + 4$ ?
Is $x + 3$ a factor of $x^{3} + 2 x^{2} - 2 x + 7$ ?
Is $x - 2$ a factor of $x^{3} + 2 x^{2} - 12 x + 14$ ?
Is $x - 5$ a factor of $x^{3} - 9 x^{2} + 23 x - 19$ ?



Sigma Prep · sigmaprep.io/worksheets Name The Remainder TheoremAnswer keyVersion 2 Date Period Find each remainder, or answer as asked. Find the remainder Find the remainder when $- 12 r^{4} - 12 r^{3} + 11 r^{2} + 12 r - 10$ is divided by $r + 2$ . $- 86$ Find the remainder when $10 x^{4} - 11 x^{3} - 12 x^{2} - 10 x + 10$ is divided by $x + 2$ . $230$ Find the remainder when $12 a^{4} - 12 a^{3} - 12 a^{2} + 12 a - 12$ is divided by $a - 2$ . $60$ Find the remainder when $- 11 m^{4} + 11 m^{3} + 12 m^{2} - 11 m - 12$ is divided by $m - 3$ . $- 531$ Find the remainder when $10 b^{4} + 11 b^{3} - 11 b^{2} + 11 b + 10$ is divided by $b + 3$ . $391$ Find the remainder when $11 p^{4} - 12 p^{3} + 10 p^{2} - 11 p - 11$ is divided by $p + 1$ . $33$ Find the remainder when $10 k^{4} + 11 k^{3} + 11 k^{2} - 12 k - 10$ is divided by $k - 3$ . $1160$ Find the remainder when $- 11 y^{4} + 12 y^{3} - 11 y^{2} + 11 y - 11$ is divided by $y - 1$ . $- 10$ Find the remainder when $- 11 p^{4} - 11 p^{3} + 10 p^{2} - 12 p + 12$ is divided by $p - 1$ . $- 12$ Find the remainder when $10 x^{4} + 11 x^{3} - 12 x^{2} + 11 x - 10$ is divided by $x + 3$ . $362$ Decide whether it is a factor Is $x - 3$ a factor of $x^{3} - x^{2} - x - 15$ ?Yes: $f (3) = 0$ Is $x + 5$ a factor of $x^{3} + 8 x^{2} + 12 x - 15$ ?Yes: $f (- 5) = 0$ Is $x + 5$ a factor of $x^{3} + 7 x^{2} + 14 x + 25$ ?No: $f (- 5) = 5$ Is $x - 2$ a factor of $x^{3} - 5 x^{2} + x + 10$ ?Yes: $f (2) = 0$ Is $x - 3$ a factor of $x^{3} - 9 x - 8$ ?No: $f (3) = - 8$ Is $x + 1$ a factor of $x^{3} + 3 x^{2} - 2$ ?Yes: $f (- 1) = 0$ Is $x - 1$ a factor of $x^{3} - 6 x^{2} + 4$ ?No: $f (1) = - 1$ Is $x + 3$ a factor of $x^{3} + 2 x^{2} - 2 x + 7$ ?No: $f (- 3) = 4$ Is $x - 2$ a factor of $x^{3} + 2 x^{2} - 12 x + 14$ ?No: $f (2) = 6$ Is $x - 5$ a factor of $x^{3} - 9 x^{2} + 23 x - 19$ ?No: $f (5) = - 4$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Remainder TheoremAnswer keyVersion 2

Date Period

Find each remainder, or answer as asked.

Find the remainder
Find the remainder when $- 12 r^{4} - 12 r^{3} + 11 r^{2} + 12 r - 10$ is divided by $r + 2$ . $- 86$
Find the remainder when $10 x^{4} - 11 x^{3} - 12 x^{2} - 10 x + 10$ is divided by $x + 2$ . $230$
Find the remainder when $12 a^{4} - 12 a^{3} - 12 a^{2} + 12 a - 12$ is divided by $a - 2$ . $60$
Find the remainder when $- 11 m^{4} + 11 m^{3} + 12 m^{2} - 11 m - 12$ is divided by $m - 3$ . $- 531$
Find the remainder when $10 b^{4} + 11 b^{3} - 11 b^{2} + 11 b + 10$ is divided by $b + 3$ . $391$
Find the remainder when $11 p^{4} - 12 p^{3} + 10 p^{2} - 11 p - 11$ is divided by $p + 1$ . $33$
Find the remainder when $10 k^{4} + 11 k^{3} + 11 k^{2} - 12 k - 10$ is divided by $k - 3$ . $1160$
Find the remainder when $- 11 y^{4} + 12 y^{3} - 11 y^{2} + 11 y - 11$ is divided by $y - 1$ . $- 10$
Find the remainder when $- 11 p^{4} - 11 p^{3} + 10 p^{2} - 12 p + 12$ is divided by $p - 1$ . $- 12$
Find the remainder when $10 x^{4} + 11 x^{3} - 12 x^{2} + 11 x - 10$ is divided by $x + 3$ . $362$

Decide whether it is a factor
Is $x - 3$ a factor of $x^{3} - x^{2} - x - 15$ ?Yes: $f (3) = 0$
Is $x + 5$ a factor of $x^{3} + 8 x^{2} + 12 x - 15$ ?Yes: $f (- 5) = 0$
Is $x + 5$ a factor of $x^{3} + 7 x^{2} + 14 x + 25$ ?No: $f (- 5) = 5$
Is $x - 2$ a factor of $x^{3} - 5 x^{2} + x + 10$ ?Yes: $f (2) = 0$
Is $x - 3$ a factor of $x^{3} - 9 x - 8$ ?No: $f (3) = - 8$
Is $x + 1$ a factor of $x^{3} + 3 x^{2} - 2$ ?Yes: $f (- 1) = 0$
Is $x - 1$ a factor of $x^{3} - 6 x^{2} + 4$ ?No: $f (1) = - 1$
Is $x + 3$ a factor of $x^{3} + 2 x^{2} - 2 x + 7$ ?No: $f (- 3) = 4$
Is $x - 2$ a factor of $x^{3} + 2 x^{2} - 12 x + 14$ ?No: $f (2) = 6$
Is $x - 5$ a factor of $x^{3} - 9 x^{2} + 23 x - 19$ ?No: $f (5) = - 4$

How to do these

Find the remainder when $x^{4} - 3 x^{3} + 2 x - 5$ is divided by $x + 2$

x + 2 is x - (-2), so c = -2.
f(-2) = 16 - 3(-8) + 2(-2) - 5 = 16 + 24 - 4 - 5.

The answer is $31$ .

Where students go wrong

Skipping a missing power in synthetic division. x⁴ - 3x³ + 2x - 5 has no x² term, so a 0 must hold its place in the row of coefficients, or every number after it is off.

Also called synthetic substitution, evaluating a polynomial or the factor theorem.

What you can put on this worksheet

The remainder on dividing by x - c: Find the remainder when $2 v^{2} - 3 v + 5$ is divided by $v - 3$ . → $14$
Is it a factor?: Is $x - 3$ a factor of $x^{3} - 6 x^{2} + 10 x - 12$ ? → No: $f (3) = - 9$
Degree five, large c, is it a factor: Is $(m - 3)$ a factor of $m^{5} + 9 m^{3} - 9 m - 5$ ? → No, the remainder is $454$
Upper and lower bound test: Use synthetic division to decide whether $- 4$ is an upper bound, a lower bound, or neither for the real zeros of $f (x) = 2 x^{3} - 5 x^{2} + 9 x - 3$ → Bottom row $2, - 13, 61, - 247$ ; $- 4$ is a lower bound
An interval that holds every real zero: Use the bound tests to find the integers $a$ and $b$ , as close together as the tests allow, so that every real zero of $f (x) = x^{3} - 3 x^{2} + 2 x$ is in $[a, b]$ . → $[0, 3]$

Questions about these worksheets

Yes. Every remainder 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 remainder 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 5: the remainder on dividing by x - c, is it a factor?, degree five, large c, is it a factor, upper and lower bound test and an interval that holds every real zero. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the wrong sign for c. Dividing by x + 4 means c is -4, because x + 4 is x - (-4). Substituting 4 gives a number that is not the remainder and looks perfectly reasonable.

Remainder theorem worksheet for Precalculus

Select your difficulty

Find the remainder

Decide whether it is a factor

Find the remainder

Decide whether it is a factor

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these remainder theorem worksheets free?+

Do the remainder theorem worksheets come with an answer key?+

Are these remainder theorem worksheets printable?+

What grade are these remainder theorem worksheets for?+

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

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

Can I choose how hard the remainder theorem questions are?+

What kinds of question are on a remainder theorem worksheet?+

What do students get wrong with remainder theorem?+