The Binomial Theorem Lesson

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 x^{n - r}y^r 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 k^th Term of a Binomial Expansion

$$(x + y)^n$$ n is a natural number.
In some cases, we will be asked to find the k^th 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 4^th term in the expansion. $$(y + 2)^5$$ Let's plug into our formula. Here k is 4 since we want the 4^th term. n is 5 since our binomial is raised to the 5^th power. $${n \choose{k - 1}}x^{n - (k - 1)}y^{k - 1}$$ $${5 \choose 3}\cdot y^{5 - (4 - 1)}\cdot 2^{4 - 1}$$ $$10 \cdot y^2 \cdot 2^3$$ $$80y^2$$

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