Lesson 12.4: Binomial Theorem

New Concept

This lesson introduces powerful tools for expanding binomials like $(a+b)^n$. You'll learn to use Pascal's Triangle and the Binomial Theorem, which uses binomial coefficients $\left(\begin{array}{c} n \\ r \end{array}\right)$, to find these expansions efficiently.

What’s next

Soon, we'll dive into worked examples for expanding binomials. You'll master these methods through interactive practice and challenge problems.

Property

Patterns in the expansion of $(a + b)^n$

• The number of terms is $n + 1$.
• The first term is $a^n$ and the last term is $b^n$.
• The exponents on $a$ decrease by one on each term going left to right.
• The exponents on $b$ increase by one on each term going left to right.
• The sum of the exponents on any term is $n$.

Pascal’s Triangle
This triangle gives the coefficients of the terms when we expand binomials. Each number in the array is the sum of the two closest numbers in the row above.

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1
 1 5 10 10 5 1

Examples

• To expand $(c+d)^4$, we use the row from Pascal's Triangle that starts with 1, 4. The coefficients are 1, 4, 6, 4, 1. The expansion is $1c^4 + 4c^3d^1 + 6c^2d^2 + 4c^1d^3 + 1d^4 = c^4 + 4c^3d + 6c^2d^2 + 4cd^3 + d^4$.

Property

A binomial coefficient $\left(\begin{array}{c} n \\ r \end{array}\right)$, where $r$ and $n$ are integers with $0 \leq r \leq n$, is defined as

We read $\left(\begin{array}{c} n \\ r \end{array}\right)$ as “$n$ choose $r$” or “$n$ taken $r$ at a time”.

Properties of Binomial Coefficients

Examples

• To evaluate $\left(\begin{array}{c} 6 \\ 1 \end{array}\right)$, we use the property $\left(\begin{array}{c} n \\ 1 \end{array}\right) = n$. So, $\left(\begin{array}{c} 6 \\ 1 \end{array}\right) = 6$.

Property

For any real numbers $a$ and $b$, and positive integer $n$,

Examples

• To expand $(x+y)^3$, we use the theorem: $\left(\begin{array}{c} 3 \\ 0 \end{array}\right)x^3 + \left(\begin{array}{c} 3 \\ 1 \end{array}\right)x^2y + \left(\begin{array}{c} 3 \\ 2 \end{array}\right)xy^2 + \left(\begin{array}{c} 3 \\ 3 \end{array}\right)y^3 = 1x^3 + 3x^2y + 3xy^2 + 1y^3$.

• To expand $(a-3)^4$, we set $a=a$ and $b=-3$. The expansion is $1(a)^4 + 4(a)^3(-3) + 6(a)^2(-3)^2 + 4(a)(-3)^3 + 1(-3)^4 = a^4 - 12a^3 + 54a^2 - 108a + 81$.

Property

The $(r + 1)$st term in the expansion of $(a + b)^n$ is

Note that for the $(r + 1)$st term, the exponent of $b$ is $r$.

Examples

• To find the third term of $(a+b)^8$, we have $n=8$. For the third term, $r+1=3$, so $r=2$. The term is $\left(\begin{array}{c} 8 \\ 2 \end{array}\right)a^{8-2}b^2 = 28a^6b^2$.

• To find the fifth term of $(x-2)^6$, we have $n=6, a=x, b=-2$. For the fifth term, $r+1=5$, so $r=4$. The term is $\left(\begin{array}{c} 6 \\ 4 \end{array}\right)x^{6-4}(-2)^4 = 15x^2(16) = 240x^2$.