Remainder theorem worksheet

Dividing by x - c leaves a remainder of exactly f(c), so there is no need to do the division at all. When the remainder is zero, x - c is a factor, which is the factor theorem in one line.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name The Remainder Theorem Date Period Find each remainder. Find the remainder when $8 a^{3} - 8 a^{2} + 9 a - 8$ is divided by $a - 3$ Find the remainder when $- 9 p^{3} - 7 p^{2} - 7 p - 8$ is divided by $p + 1$ Find the remainder when $9 x^{3} - 8 x^{2} + 7 x - 8$ is divided by $x + 4$ Find the remainder when $- 7 p^{3} - 7 p^{2} - 9 p + 9$ is divided by $p + 4$ Find the remainder when $9 v^{3} + 7 v^{2} - 8 v - 7$ is divided by $v + 2$ Find the remainder when $- 7 a^{3} + 9 a^{2} + 9 a + 9$ is divided by $a - 2$ Find the remainder when $- 9 k^{3} + 9 k^{2} + 7 k + 9$ is divided by $k + 2$ Find the remainder when $9 n^{3} + 7 n^{2} + 8 n - 7$ is divided by $n + 1$ Find the remainder when $- 7 m^{3} + 7 m^{2} - 8 m + 7$ is divided by $m + 2$ Find the remainder when $- 9 a^{3} - 9 a^{2} + 8 a + 7$ is divided by $a - 5$ Find the remainder when $- 7 x^{3} + 9 x^{2} + 7 x - 7$ is divided by $x - 4$ Find the remainder when $7 r^{3} + 7 r^{2} - 8 r - 7$ is divided by $r - 5$ Find the remainder when $9 a^{3} - 7 a^{2} - 7 a + 7$ is divided by $a + 2$ Find the remainder when $- 9 x^{3} - 9 x^{2} - 7 x - 9$ is divided by $x - 3$ Find the remainder when $- 8 a^{3} + 7 a^{2} + 9 a - 7$ is divided by $a - 1$ Find the remainder when $- 9 m^{3} - 8 m^{2} + 8 m - 9$ is divided by $m - 3$ Find the remainder when $7 a^{3} + 7 a^{2} + 7 a - 7$ is divided by $a - 5$ Find the remainder when $7 x^{3} + 8 x^{2} + 7 x - 7$ is divided by $x + 5$ Find the remainder when $- 9 y^{3} + 8 y^{2} - 8 y - 9$ is divided by $y + 2$ Find the remainder when $- 8 y^{3} + 7 y^{2} - 7 y + 8$ is divided by $y - 1$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Remainder Theorem

Date Period

Find each remainder.

Find the remainder when $8 a^{3} - 8 a^{2} + 9 a - 8$ is divided by $a - 3$
Find the remainder when $- 9 p^{3} - 7 p^{2} - 7 p - 8$ is divided by $p + 1$
Find the remainder when $9 x^{3} - 8 x^{2} + 7 x - 8$ is divided by $x + 4$
Find the remainder when $- 7 p^{3} - 7 p^{2} - 9 p + 9$ is divided by $p + 4$
Find the remainder when $9 v^{3} + 7 v^{2} - 8 v - 7$ is divided by $v + 2$
Find the remainder when $- 7 a^{3} + 9 a^{2} + 9 a + 9$ is divided by $a - 2$
Find the remainder when $- 9 k^{3} + 9 k^{2} + 7 k + 9$ is divided by $k + 2$
Find the remainder when $9 n^{3} + 7 n^{2} + 8 n - 7$ is divided by $n + 1$
Find the remainder when $- 7 m^{3} + 7 m^{2} - 8 m + 7$ is divided by $m + 2$
Find the remainder when $- 9 a^{3} - 9 a^{2} + 8 a + 7$ is divided by $a - 5$
Find the remainder when $- 7 x^{3} + 9 x^{2} + 7 x - 7$ is divided by $x - 4$
Find the remainder when $7 r^{3} + 7 r^{2} - 8 r - 7$ is divided by $r - 5$
Find the remainder when $9 a^{3} - 7 a^{2} - 7 a + 7$ is divided by $a + 2$
Find the remainder when $- 9 x^{3} - 9 x^{2} - 7 x - 9$ is divided by $x - 3$
Find the remainder when $- 8 a^{3} + 7 a^{2} + 9 a - 7$ is divided by $a - 1$
Find the remainder when $- 9 m^{3} - 8 m^{2} + 8 m - 9$ is divided by $m - 3$
Find the remainder when $7 a^{3} + 7 a^{2} + 7 a - 7$ is divided by $a - 5$
Find the remainder when $7 x^{3} + 8 x^{2} + 7 x - 7$ is divided by $x + 5$
Find the remainder when $- 9 y^{3} + 8 y^{2} - 8 y - 9$ is divided by $y + 2$
Find the remainder when $- 8 y^{3} + 7 y^{2} - 7 y + 8$ is divided by $y - 1$



Sigma Prep · sigmaprep.io/worksheets Name The Remainder TheoremAnswer keyVersion 1 Date Period Find each remainder. Find the remainder when $8 a^{3} - 8 a^{2} + 9 a - 8$ is divided by $a - 3$ $163$ Find the remainder when $- 9 p^{3} - 7 p^{2} - 7 p - 8$ is divided by $p + 1$ $1$ Find the remainder when $9 x^{3} - 8 x^{2} + 7 x - 8$ is divided by $x + 4$ $- 740$ Find the remainder when $- 7 p^{3} - 7 p^{2} - 9 p + 9$ is divided by $p + 4$ $381$ Find the remainder when $9 v^{3} + 7 v^{2} - 8 v - 7$ is divided by $v + 2$ $- 35$ Find the remainder when $- 7 a^{3} + 9 a^{2} + 9 a + 9$ is divided by $a - 2$ $7$ Find the remainder when $- 9 k^{3} + 9 k^{2} + 7 k + 9$ is divided by $k + 2$ $103$ Find the remainder when $9 n^{3} + 7 n^{2} + 8 n - 7$ is divided by $n + 1$ $- 17$ Find the remainder when $- 7 m^{3} + 7 m^{2} - 8 m + 7$ is divided by $m + 2$ $107$ Find the remainder when $- 9 a^{3} - 9 a^{2} + 8 a + 7$ is divided by $a - 5$ $- 1303$ Find the remainder when $- 7 x^{3} + 9 x^{2} + 7 x - 7$ is divided by $x - 4$ $- 283$ Find the remainder when $7 r^{3} + 7 r^{2} - 8 r - 7$ is divided by $r - 5$ $1003$ Find the remainder when $9 a^{3} - 7 a^{2} - 7 a + 7$ is divided by $a + 2$ $- 79$ Find the remainder when $- 9 x^{3} - 9 x^{2} - 7 x - 9$ is divided by $x - 3$ $- 354$ Find the remainder when $- 8 a^{3} + 7 a^{2} + 9 a - 7$ is divided by $a - 1$ $1$ Find the remainder when $- 9 m^{3} - 8 m^{2} + 8 m - 9$ is divided by $m - 3$ $- 300$ Find the remainder when $7 a^{3} + 7 a^{2} + 7 a - 7$ is divided by $a - 5$ $1078$ Find the remainder when $7 x^{3} + 8 x^{2} + 7 x - 7$ is divided by $x + 5$ $- 717$ Find the remainder when $- 9 y^{3} + 8 y^{2} - 8 y - 9$ is divided by $y + 2$ $111$ Find the remainder when $- 8 y^{3} + 7 y^{2} - 7 y + 8$ is divided by $y - 1$ $0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Remainder TheoremAnswer keyVersion 1

Date Period

Find each remainder.

Find the remainder when $8 a^{3} - 8 a^{2} + 9 a - 8$ is divided by $a - 3$ $163$
Find the remainder when $- 9 p^{3} - 7 p^{2} - 7 p - 8$ is divided by $p + 1$ $1$
Find the remainder when $9 x^{3} - 8 x^{2} + 7 x - 8$ is divided by $x + 4$ $- 740$
Find the remainder when $- 7 p^{3} - 7 p^{2} - 9 p + 9$ is divided by $p + 4$ $381$
Find the remainder when $9 v^{3} + 7 v^{2} - 8 v - 7$ is divided by $v + 2$ $- 35$
Find the remainder when $- 7 a^{3} + 9 a^{2} + 9 a + 9$ is divided by $a - 2$ $7$
Find the remainder when $- 9 k^{3} + 9 k^{2} + 7 k + 9$ is divided by $k + 2$ $103$
Find the remainder when $9 n^{3} + 7 n^{2} + 8 n - 7$ is divided by $n + 1$ $- 17$
Find the remainder when $- 7 m^{3} + 7 m^{2} - 8 m + 7$ is divided by $m + 2$ $107$
Find the remainder when $- 9 a^{3} - 9 a^{2} + 8 a + 7$ is divided by $a - 5$ $- 1303$
Find the remainder when $- 7 x^{3} + 9 x^{2} + 7 x - 7$ is divided by $x - 4$ $- 283$
Find the remainder when $7 r^{3} + 7 r^{2} - 8 r - 7$ is divided by $r - 5$ $1003$
Find the remainder when $9 a^{3} - 7 a^{2} - 7 a + 7$ is divided by $a + 2$ $- 79$
Find the remainder when $- 9 x^{3} - 9 x^{2} - 7 x - 9$ is divided by $x - 3$ $- 354$
Find the remainder when $- 8 a^{3} + 7 a^{2} + 9 a - 7$ is divided by $a - 1$ $1$
Find the remainder when $- 9 m^{3} - 8 m^{2} + 8 m - 9$ is divided by $m - 3$ $- 300$
Find the remainder when $7 a^{3} + 7 a^{2} + 7 a - 7$ is divided by $a - 5$ $1078$
Find the remainder when $7 x^{3} + 8 x^{2} + 7 x - 7$ is divided by $x + 5$ $- 717$
Find the remainder when $- 9 y^{3} + 8 y^{2} - 8 y - 9$ is divided by $y + 2$ $111$
Find the remainder when $- 8 y^{3} + 7 y^{2} - 7 y + 8$ is divided by $y - 1$ $0$

How to do these

Find the remainder when $2 x^{3} - 5 x + 1$ is divided by $x - 3$

The divisor is x - 3, so c is 3.
Work out f(3): 2(27) - 5(3) + 1.

The answer is $40$ .

Where students go wrong

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.

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

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.

Algebra 2 is usually taken around 10th or 11th 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.

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

Select your difficulty

The Remainder Theorem

The Remainder TheoremAnswer keyVersion 1

How to do these

Where students go wrong

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 do students get wrong with remainder theorem?+