Lesson Objectives
• Learn how to use factorial notation
• Learn how to evaluate a combination
• Learn how to find the expansion of a binomial
• Learn how to find the kth term of a binomial expansion

## How to Use the Binomial Theorem

In this lesson, we will learn about the binomial theorem. This theorem allows us to quickly expand a binomial of the form: $$(x + y)^n$$ Where n is a natural number.

### Factorial Notation

Before we get into the binomial theorem, we need to learn about factorial notation. When we see something such as: $$3!$$ This is telling us to start with 3 then multiply by 1 less or 2, then multiply by 1 less or 1. $$3!=3 \cdot 2 \cdot 1=6$$ In other words, we start with the whole number given and multiply by each whole number going down to 1. Let's look at an example.
Example #1: Evaluate each. $$6!$$ $$6!=6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=720$$

### Binomial Coefficients Formula

When we work with the binomial theorem, we will need to understand how to calculate a number known as a binomial coefficient. This is done based on the following formula: $$\hspace{.2em}{n \choose r}=\frac{n!}{r!(n - r)!}$$ This is read as n choose r. Let's look at an example.
Example #2: Evaluate each. $$\hspace{.2em}{8 \choose 5}$$ Let's plug into our formula: $$\hspace{.2em}{8 \choose 5}=\frac{8!}{5!(8 - 5)!}$$ $$\hspace{.2em}{8 \choose 5}=\frac{8!}{5!3!}$$ $$\hspace{.2em}{8 \choose 5}=\frac{40{,}320}{(120)(6)}$$ $$\hspace{.2em}{8 \choose 5}=\frac{40{,}320}{720}$$ $$\hspace{.2em}{8 \choose 5}=56$$

### The Binomial Theorem

$$(x + y)^n$$ n is a natural number.
• There will be n + 1 terms
• The first term is xn
• The last term is yn
• In each succeeding term, the exponent on x decreases by 1
• In each succeeding term, the exponent on y increases by 1
• The sum of the exponents on x and y in any term is always n
• The coefficient of the term with xn - ryr is n choose r
Let's look at an example.
Example #3: Find the expansion. $$(x + 2)^4$$ There will be a total of 5 terms since n is 4.
The first term is: $$x^4$$ The last term or fifth term is: $$2^4=16$$ To find the second term, the exponent on x decreases by 1, the exponent on 2 increases by 1, the coefficient is found as 4 choose 1. This gives us: $$2^1 \cdot{4 \choose 1}\cdot x^3$$ $$2 \cdot 4 \cdot x^3$$ $$8x^3$$ To find the third term, the exponent on x decreases by 1, the exponent on 2 increases by 1, the coefficient is found as 4 choose 2. This gives us: $$2^2 \cdot{4 \choose 2}\cdot x^2$$ $$4 \cdot 6 \cdot x^2$$ $$24x^2$$ To find the fourth term, the exponent on x decreases by 1, the exponent on 2 increases by 1, the coefficient is found as 4 choose 3. $$2^3 \cdot{4 \choose 3}\cdot x$$ $$8 \cdot 4 \cdot x$$ $$32x$$ Now we can put our terms together: $$x^4 + 8x^3 + 24x^2 + 32x + 16$$

### Finding the kth Term of a Binomial Expansion

$$(x + y)^n$$ n is a natural number.
In some cases, we will be asked to find the kth term of a binomial expansion. Instead of writing the entire expansion out and then picking out the term, we can use the following formula: $${n \choose{k - 1}}x^{n - (k - 1)}y^{k - 1}$$ Let's look at an example.
Example #4: Find the 4th term in the expansion. $$(y + 2)^5$$ Let's plug into our formula. Here k is 4 since we want the 4th term. n is 5 since our binomial is raised to the 5th power. $${n \choose{k - 1}}x^{n - (k - 1)}y^{k - 1}$$ $${5 \choose 3}\cdot y^{5 - (4 - 1)}\cdot 2^{4 - 1}$$ $$10 \cdot y^2 \cdot 2^3$$ $$80y^2$$

#### Skills Check:

Example #1

Find each term described.

4th term in expansion of:

$$(x + 2)^5$$

A
$$16x^4$$
B
$$80x^2$$
C
$$40x^3$$
D
$$40x^2$$
E
$$4x^2$$

Example #2

Find each term described.

6th term in expansion of:

$$(2y + 1)^5$$

A
$$1$$
B
$$10y$$
C
$$32y^5$$
D
$$5y^4$$
E
$$2y^3$$

Example #3

Find each term described.

3rd term in expansion of:

$$(x + 4)^4$$

A
$$1024x^4$$
B
$$16x$$
C
$$16x^3$$
D
$$96x^2$$
E
$$256x$$