Precalculus The Binomial Theorem Worksheet (PDF)

Precalculus math worksheet: The Binomial Theorem – Maths PDF

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This binomial theorem worksheet for Precalculus covers the binomial theorem (Common Core A-APR.C.5), with a free PDF to print.

These 20 problems practice the Binomial Theorem (Common Core A-APR.C.5). You will use factorials and binomial coefficients, read Pascal’s triangle, expand binomials, find single terms and solve counting and probability problems.

Precalculus · Common Core State Standards: A-APR.C.5

Binomial theorem worksheet: exercises 1 to 10

Exercise 1 – Factorials

Evaluate \(6!\), \(0!\) and \(\frac{8!}{6!}\).

Exercise 2 – Binomial coefficients

Evaluate \(C(7,3)\), \(C(10,2)\), \(C(9,9)\) and \(C(12,10)\).

Exercise 3 – Pascal's triangle

Rows 0 through 5.

Use the triangle to write row 6.

Exercise 4 – Expand (x + y)^4

Expand \((x+y)^{4}\).

Exercise 5 – Expand (x + 2)^5

Terms of (x + 2)^5.

Expand \((x+2)^{5}\).

Exercise 6 – Expand (3x + 1)^4

Expand \((3x+1)^{4}\).

Exercise 7 – Expand (x – 3)^3

Expand \((x-3)^{3}\).

Exercise 8 – Find a single term

Find the 4th term of \((x+3)^{8}\).

Exercise 9 – Find a coefficient

Find the coefficient of \(x^{3}\) in \((2x+1)^{6}\).

Exercise 10 – Coefficient with a negative

Find the coefficient of \(x^{4}\) in \((x-2)^{7}\).

Exercises 11 and up

Exercise 11 – Constant term

Find the constant term in \(\left(x+\frac{2}{x}\right)^{6}\).

Exercise 12 – Sum of coefficients

Find the sum of the coefficients of \((3x-1)^{5}\).

Exercise 13 – Row sums

Entries of row 5.

Add the entries of row 5 of Pascal’s triangle and compare with \(2^{5}\).

Exercise 14 – Middle term

Find the middle term of \((x+y)^{10}\).

Exercise 15 – Approximate a power

Use the first four terms of \((1+0.02)^{5}\) to approximate \(1.02^{5}\). Compare with a calculator value \(1.10408\).

Exercise 16 – Complex number

Use the Binomial Theorem to find \((1+i)^{4}\).

Exercise 17 – Choose a committee

A teacher chooses \(3\) of \(8\) students for a committee. How many different committees are possible?

Exercise 18 – Coin tosses

Row 5 counts outcomes of 5 tosses.

Find the probability of exactly \(3\) heads in \(5\) tosses of a fair coin.

Exercise 19 – Free throws

Probability of k made shots out of 5.

Mei makes \(70\%\) of her free throws. Find the probability that she makes exactly \(4\) of \(5\) shots.

Exercise 20 – Two variables

Expand \((2x-3y)^{4}\).

Key facts behind this binomial theorem worksheet

Key facts

n! = n(n – 1)…(1) and 0! = 1; C(n, k) = n!/(k!(n – k)!).
Pascal’s triangle: each entry is the sum of the two above; row n sums to 2^n.
(a + b)^n = sum of C(n, k) a^(n-k) b^k for k = 0 to n.
The general term is T(k+1) = C(n, k) a^(n-k) b^k; signs alternate when b is negative.
Setting x = 1 gives the sum of the coefficients.

How to use this binomial theorem worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

All Precalculus chapters: Precalculus math.

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