Precalculus The Binomial Theorem Lesson (Free PDF)

Precalculus math lesson: The Binomial Theorem – Maths PDF

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

The Binomial Theorem expands (a + b)^n as the sum of C(n, k) a^(n-k) b^k for k from 0 to n, where C(n, k) = n!/(k!(n – k)!) counts the ways to choose k items from n. The coefficients are the entries in row n of Pascal’s triangle, and they add up to 2^n.

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

Key facts in this binomial theorem lesson

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.

What are factorials and binomial coefficients?

The factorial \(n!\) is the product \(n(n-1)(n-2)\text{ … }(1)\), with \(0!=1\). The binomial coefficient \(C(n,k)=\frac{n!}{k!\,(n-k)!}\) counts the ways to choose \(k\) items from \(n\) when order does not matter. For example \(C(5,2)=\frac{5!}{2!\,3!}=\frac{120}{12}=10\). Two useful facts are \(C(n,0)=C(n,n)=1\) and \(C(n,k)=C(n,n-k)\).

What is Pascal’s triangle?

Pascal’s triangle lists the binomial coefficients: row \(n\) contains \(C(n,0),C(n,1),\text{ … },C(n,n)\). Each entry is the sum of the two entries above it, because \(C(n,k)=C(n-1,k-1)+C(n-1,k)\). The entries in row \(n\) add up to \(2^{n}\).

Pascal’s triangle. Row 5 (shaded) is 1, 5, 10, 10, 5, 1, and its sum is 32 = 2^5.

Look at the shape of each row. The entries rise to the middle and fall again, and the row reads the same forward and backward, which is the symmetry \(C(n,k)=C(n,n-k)\). This symmetry is useful: you only need to compute half of a row, and you can check an expansion by seeing that the coefficients of the first and last terms match. The rows also contain surprises, such as the powers of 11 (row 4 gives \(1,4,6,4,1\), and \(11^{4}=14641\)).

The entries of row 6 are symmetric: 1, 6, 15, 20, 15, 6, 1.

How do you expand a power of a binomial?

Binomial Theorem

\((a+b)^{n}=\sum_{k=0}^{n}C(n,k)\,a^{n-k}b^{k}\). The terms have exponents of \(a\) that decrease from \(n\) to 0 and exponents of \(b\) that increase from 0 to \(n\).

Squaring shows the idea geometrically: \((a+b)^{2}=a^{2}+2ab+b^{2}\), where the 2 is the coefficient \(C(2,1)\).

A square with side a + b splits into regions a², ab, ab and b².

Example: Expand \((x+2)^{5}\). Use row 5 of Pascal’s triangle and powers of 2.

Each term is \(C(5,k)\cdot x^{5-k}\cdot2^{k}\). The sum is \(x^{5}+10x^{4}+40x^{3}+80x^{2}+80x+32\).

When the second term is negative, the signs alternate: \((x-3)^{3}=x^{3}-9x^{2}+27x-27\). To find a single term, use the general term \(T_{k+1}=C(n,k)\,a^{n-k}b^{k}\) instead of expanding everything. The sum of all coefficients of \((ax+b)^{n}\) is found by setting \(x=1\), giving \((a+b)^{n}\). Binomial coefficients also solve counting and probability problems: the number of ways to get exactly 3 heads in 5 tosses is \(C(5,3)=10\).

Binomial theorem lesson: questions and answers

What does C(n, k) mean?

It is the number of ways to choose k objects from n when order does not matter, read ‘n choose k’. For example C(5, 2) = 10.

How does Pascal’s triangle give the coefficients?

Row n lists C(n, 0) through C(n, n). Row 4 is 1, 4, 6, 4, 1, so (a + b)^4 = a^4 + 4a^3 b + 6a^2 b^2 + 4a b^3 + b^4.

How many terms are in the expansion of (a + b)^n?

There are n + 1 terms, one for each value of k from 0 to n.

How do I find one term without expanding?

Use the general term C(n, k) a^(n-k) b^k with the k that gives the power you want. The (k + 1)th term uses k.

Why do the signs alternate for (x – 3)^n?

Because b is negative, b^k is negative for odd k and positive for even k, so the terms alternate in sign.

How to use this binomial theorem lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the binomial theorem lesson as a PDF with the button below and keep it with your notes.

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