1024a^5 + 14080a^4 + 77440a^3 + 212960a^2 + 292820a + 161051

4096b^6 + 30720b^5 + 96000b^4 + 160000b^3 + 150000b^2 + 75000b + 15625

32p^5 - 400p^4 + 2000p^3 - 5000p^2 + 6250p - 3125

1024n^5 - 6400n^4 + 16000n^3 - 20000n^2 + 12500n - 3125

243b^5 - 2025b^4 + 6750b^3 - 11250b^2 + 9375b - 3125

243r^5 - 3240r^4 + 17280r^3 - 46080r^2 + 61440r - 32768

1024r^5 + 6400r^4 + 16000r^3 + 20000r^2 + 12500r + 3125

32y^5 + 160y^4 + 320y^3 + 320y^2 + 160y + 32

243p^5 - 3240p^4 + 17280p^3 - 46080p^2 + 61440p - 32768

1024n^5 - 8960n^4 + 31360n^3 - 54880n^2 + 48020n - 16807

64b^6 + 960b^5 + 6000b^4 + 20000b^3 + 37500b^2 + 37500b + 15625

4096a^6 + 49152a^5 + 245760a^4 + 655360a^3 + 983040a^2 + 786432a + 262144

4096k^6 - 6144k^5 + 3840k^4 - 1280k^3 + 240k^2 - 24k + 1

4096b^6 + 67584b^5 + 464640b^4 + 1703680b^3 + 3513840b^2 + 3865224b + 1771561

4096n^6 + 12288n^5 + 15360n^4 + 10240n^3 + 3840n^2 + 768n + 64

32n^5 - 400n^4 + 2000n^3 - 5000n^2 + 6250n - 3125

4096a^6 + 6144a^5 + 3840a^4 + 1280a^3 + 240a^2 + 24a + 1

1024k^5 - 5120k^4 + 10240k^3 - 10240k^2 + 5120k - 1024

32b^5 + 560b^4 + 3920b^3 + 13720b^2 + 24010b + 16807

1024n^5 - 5120n^4 + 10240n^3 - 10240n^2 + 5120n - 1024

Worksheets · Precalculus

Precalculus binomial theorem worksheet

The coefficients come from Pascal's triangle and the powers march in opposite directions. Once the pattern is visible, a fifth power is no harder than a third.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name The Binomial Theorem Date Period Expand each expression. ${(4 a + 11)}^{5}$ ${(4 b + 5)}^{6}$ ${(2 p - 5)}^{5}$ ${(4 n - 5)}^{5}$ ${(3 b - 5)}^{5}$ ${(3 r - 8)}^{5}$ ${(4 r + 5)}^{5}$ ${(2 y + 2)}^{5}$ ${(3 p - 8)}^{5}$ ${(4 n - 7)}^{5}$ ${(2 b + 5)}^{6}$ ${(4 a + 8)}^{6}$ ${(4 k - 1)}^{6}$ ${(4 b + 11)}^{6}$ ${(4 n + 2)}^{6}$ ${(2 n - 5)}^{5}$ ${(4 a + 1)}^{6}$ ${(4 k - 4)}^{5}$ ${(2 b + 7)}^{5}$ ${(4 n - 4)}^{5}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Binomial Theorem

Date Period

Expand each expression.

${(4 a + 11)}^{5}$
${(4 b + 5)}^{6}$
${(2 p - 5)}^{5}$
${(4 n - 5)}^{5}$
${(3 b - 5)}^{5}$
${(3 r - 8)}^{5}$
${(4 r + 5)}^{5}$
${(2 y + 2)}^{5}$
${(3 p - 8)}^{5}$
${(4 n - 7)}^{5}$
${(2 b + 5)}^{6}$
${(4 a + 8)}^{6}$
${(4 k - 1)}^{6}$
${(4 b + 11)}^{6}$
${(4 n + 2)}^{6}$
${(2 n - 5)}^{5}$
${(4 a + 1)}^{6}$
${(4 k - 4)}^{5}$
${(2 b + 7)}^{5}$
${(4 n - 4)}^{5}$



Sigma Prep · sigmaprep.io/worksheets Name The Binomial TheoremAnswer keyVersion 1 Date Period Expand each expression. ${(4 a + 11)}^{5}$ $1024 a^{5} + 14080 a^{4} + 77440 a^{3} + 212960 a^{2} + 292820 a + 161051$ ${(4 b + 5)}^{6}$ $4096 b^{6} + 30720 b^{5} + 96000 b^{4} + 160000 b^{3} + 150000 b^{2} + 75000 b + 15625$ ${(2 p - 5)}^{5}$ $32 p^{5} - 400 p^{4} + 2000 p^{3} - 5000 p^{2} + 6250 p - 3125$ ${(4 n - 5)}^{5}$ $1024 n^{5} - 6400 n^{4} + 16000 n^{3} - 20000 n^{2} + 12500 n - 3125$ ${(3 b - 5)}^{5}$ $243 b^{5} - 2025 b^{4} + 6750 b^{3} - 11250 b^{2} + 9375 b - 3125$ ${(3 r - 8)}^{5}$ $243 r^{5} - 3240 r^{4} + 17280 r^{3} - 46080 r^{2} + 61440 r - 32768$ ${(4 r + 5)}^{5}$ $1024 r^{5} + 6400 r^{4} + 16000 r^{3} + 20000 r^{2} + 12500 r + 3125$ ${(2 y + 2)}^{5}$ $32 y^{5} + 160 y^{4} + 320 y^{3} + 320 y^{2} + 160 y + 32$ ${(3 p - 8)}^{5}$ $243 p^{5} - 3240 p^{4} + 17280 p^{3} - 46080 p^{2} + 61440 p - 32768$ ${(4 n - 7)}^{5}$ $1024 n^{5} - 8960 n^{4} + 31360 n^{3} - 54880 n^{2} + 48020 n - 16807$ ${(2 b + 5)}^{6}$ $64 b^{6} + 960 b^{5} + 6000 b^{4} + 20000 b^{3} + 37500 b^{2} + 37500 b + 15625$ ${(4 a + 8)}^{6}$ $4096 a^{6} + 49152 a^{5} + 245760 a^{4} + 655360 a^{3} + 983040 a^{2} + 786432 a + 262144$ ${(4 k - 1)}^{6}$ $4096 k^{6} - 6144 k^{5} + 3840 k^{4} - 1280 k^{3} + 240 k^{2} - 24 k + 1$ ${(4 b + 11)}^{6}$ $4096 b^{6} + 67584 b^{5} + 464640 b^{4} + 1703680 b^{3} + 3513840 b^{2} + 3865224 b + 1771561$ ${(4 n + 2)}^{6}$ $4096 n^{6} + 12288 n^{5} + 15360 n^{4} + 10240 n^{3} + 3840 n^{2} + 768 n + 64$ ${(2 n - 5)}^{5}$ $32 n^{5} - 400 n^{4} + 2000 n^{3} - 5000 n^{2} + 6250 n - 3125$ ${(4 a + 1)}^{6}$ $4096 a^{6} + 6144 a^{5} + 3840 a^{4} + 1280 a^{3} + 240 a^{2} + 24 a + 1$ ${(4 k - 4)}^{5}$ $1024 k^{5} - 5120 k^{4} + 10240 k^{3} - 10240 k^{2} + 5120 k - 1024$ ${(2 b + 7)}^{5}$ $32 b^{5} + 560 b^{4} + 3920 b^{3} + 13720 b^{2} + 24010 b + 16807$ ${(4 n - 4)}^{5}$ $1024 n^{5} - 5120 n^{4} + 10240 n^{3} - 10240 n^{2} + 5120 n - 1024$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Binomial TheoremAnswer keyVersion 1

Date Period

Expand each expression.

${(4 a + 11)}^{5}$ $1024 a^{5} + 14080 a^{4} + 77440 a^{3} + 212960 a^{2} + 292820 a + 161051$
${(4 b + 5)}^{6}$ $4096 b^{6} + 30720 b^{5} + 96000 b^{4} + 160000 b^{3} + 150000 b^{2} + 75000 b + 15625$
${(2 p - 5)}^{5}$ $32 p^{5} - 400 p^{4} + 2000 p^{3} - 5000 p^{2} + 6250 p - 3125$
${(4 n - 5)}^{5}$ $1024 n^{5} - 6400 n^{4} + 16000 n^{3} - 20000 n^{2} + 12500 n - 3125$
${(3 b - 5)}^{5}$ $243 b^{5} - 2025 b^{4} + 6750 b^{3} - 11250 b^{2} + 9375 b - 3125$
${(3 r - 8)}^{5}$ $243 r^{5} - 3240 r^{4} + 17280 r^{3} - 46080 r^{2} + 61440 r - 32768$
${(4 r + 5)}^{5}$ $1024 r^{5} + 6400 r^{4} + 16000 r^{3} + 20000 r^{2} + 12500 r + 3125$
${(2 y + 2)}^{5}$ $32 y^{5} + 160 y^{4} + 320 y^{3} + 320 y^{2} + 160 y + 32$
${(3 p - 8)}^{5}$ $243 p^{5} - 3240 p^{4} + 17280 p^{3} - 46080 p^{2} + 61440 p - 32768$
${(4 n - 7)}^{5}$ $1024 n^{5} - 8960 n^{4} + 31360 n^{3} - 54880 n^{2} + 48020 n - 16807$
${(2 b + 5)}^{6}$ $64 b^{6} + 960 b^{5} + 6000 b^{4} + 20000 b^{3} + 37500 b^{2} + 37500 b + 15625$
${(4 a + 8)}^{6}$ $4096 a^{6} + 49152 a^{5} + 245760 a^{4} + 655360 a^{3} + 983040 a^{2} + 786432 a + 262144$
${(4 k - 1)}^{6}$ $4096 k^{6} - 6144 k^{5} + 3840 k^{4} - 1280 k^{3} + 240 k^{2} - 24 k + 1$
${(4 b + 11)}^{6}$ $4096 b^{6} + 67584 b^{5} + 464640 b^{4} + 1703680 b^{3} + 3513840 b^{2} + 3865224 b + 1771561$
${(4 n + 2)}^{6}$ $4096 n^{6} + 12288 n^{5} + 15360 n^{4} + 10240 n^{3} + 3840 n^{2} + 768 n + 64$
${(2 n - 5)}^{5}$ $32 n^{5} - 400 n^{4} + 2000 n^{3} - 5000 n^{2} + 6250 n - 3125$
${(4 a + 1)}^{6}$ $4096 a^{6} + 6144 a^{5} + 3840 a^{4} + 1280 a^{3} + 240 a^{2} + 24 a + 1$
${(4 k - 4)}^{5}$ $1024 k^{5} - 5120 k^{4} + 10240 k^{3} - 10240 k^{2} + 5120 k - 1024$
${(2 b + 7)}^{5}$ $32 b^{5} + 560 b^{4} + 3920 b^{3} + 13720 b^{2} + 24010 b + 16807$
${(4 n - 4)}^{5}$ $1024 n^{5} - 5120 n^{4} + 10240 n^{3} - 10240 n^{2} + 5120 n - 1024$

How to do these

${(x + 2)}^{3}$

The coefficients for the third power are 1, 3, 3, 1.
The powers of x go 3, 2, 1, 0 while the 2s go 0, 1, 2, 3.
So x cubed, plus 3x squared times 2, plus 3x times 4, plus 8.

The answer is $x^{3} + 6 x^{2} + 12 x + 8$ .

Where students go wrong

Forgetting to raise the second term to its power as well. In (x + 2) cubed the last term is 2 cubed, which is 8, not 2. Every term in the parenthesis takes the power, not just the letter.

Also called binomial expansion, Pascal's triangle or expanding parentheses to a power.

Questions about these worksheets

Yes. Every binomial 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 binomial 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 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.

Forgetting to raise the second term to its power as well. In (x + 2) cubed the last term is 2 cubed, which is 8, not 2. Every term in the parenthesis takes the power, not just the letter.

Precalculus binomial theorem worksheet

Select your difficulty

The Binomial Theorem

The Binomial TheoremAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

Are these binomial theorem worksheets free?+

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

Are these binomial theorem worksheets printable?+

What grade are these binomial theorem worksheets for?+

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

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

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