Precalculus The Binomial Theorem Test with Point Values (PDF)

Precalculus math test: The Binomial Theorem – Maths PDF

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

This 20-point test covers binomial coefficients, Pascal’s triangle, expansions, coefficients and probability with coin tosses (A-APR.C.5). You have 45 minutes. Show your work.

Time and points

About this test

Grade: Precalculus
Chapter: The Binomial Theorem
Standards: A-APR.C.5
Time: 45 minutes
Total: 20 points

Binomial theorem test: questions

Question 1 – Evaluate [2 pts]

Evaluate \(C(9,4)\) and \(\frac{7!}{4!\,3!}\).

Question 2 – Expand [2 pts]

Pascal’s triangle, rows 0 to 5.

Expand \((x+1)^{5}\).

Question 3 – Expand with a coefficient [2 pts]

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

Question 4 – A single term [2 pts]

Find the 3rd term of \((x+4)^{6}\).

Question 5 – Pascal's triangle [3 pts]

Pascal’s triangle, rows 0 to 5.

(a) Use row 4 to list the coefficients of \((a+b)^{4}\). (b) Expand \((x+2)^{4}\). (c) Find the sum of the coefficients of \((x+2)^{4}\).

Question 6 – A coefficient [2 pts]

Find the coefficient of \(x^{2}\) in \((3x-2)^{5}\).

Question 7 – Constant term [2 pts]

Find the constant term of \(\left(x^{2}+\frac1x\right)^{9}\).

Question 8 – Probability [3 pts]

Row 6: ways to get k heads in 6 tosses.

A fair coin is tossed \(6\) times. (a) How many outcomes have exactly \(2\) heads? (b) How many outcomes are there in total? (c) Find the probability of exactly \(2\) heads to three decimal places.

Question 9 – Approximation [2 pts]

Use the first three terms of \((1-0.02)^{4}\) to approximate \(0.98^{4}\).

Key facts behind this binomial theorem test

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 test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Precalculus chapters: Precalculus math.

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