91 Binomial Theorem
Learning Objectives
In this section, you will:
- Apply the Binomial Theorem.
A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find[latex]\,{\left(x+y\right)}^{n}\,[/latex]without multiplying the binomial by itself [latex]n[/latex] times.
Identifying Binomial Coefficients
In Counting Principles, we studied combinations. In the shortcut to finding[latex]\,{\left(x+y\right)}^{n},\,[/latex]we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation[latex]\,\left(\begin{array}{c}n\\ r\end{array}\right)\,[/latex] instead of [latex]C\left(n,r\right),[/latex] but it can be calculated in the same way. So
The combination[latex]\,\left(\begin{array}{c}n\\ r\end{array}\right)\,[/latex]is called a binomial coefficient. An example of a binomial coefficient is[latex]\,\left(\begin{array}{c}5\\ 2\end{array}\right)=C\left(5,2\right)=10.\,[/latex]
Binomial Coefficients
If [latex]n[/latex] and [latex]r[/latex]are integers greater than or equal to 0 with [latex]n\ge r,[/latex] then the binomial coefficient is
Is a binomial coefficient always a whole number?
Yes. Just as the number of combinations must always be a whole number, a binomial coefficient will always be a whole number.
Finding Binomial Coefficients
Find each binomial coefficient.
- [latex]\left(\begin{array}{c}5\\ 3\end{array}\right)[/latex]
- [latex]\left(\begin{array}{c}9\\ 2\end{array}\right)[/latex]
- [latex]\left(\begin{array}{c}9\\ 7\end{array}\right)[/latex]
[hidden-answer a=”fs-id1165137933188″]Use the formula to calculate each binomial coefficient. You can also use the [latex]{n}_{}{C}_{r}[/latex] function on your calculator.
- [latex]\left(\begin{array}{c}5\\ 3\end{array}\right)=\frac{5!}{3!\left(5-3\right)!}=\frac{5\cdot 4\cdot 3!}{3!2!}=10[/latex]
- [latex]\left(\begin{array}{c}9\\ 2\end{array}\right)=\frac{9!}{2!\left(9-2\right)!}=\frac{9\cdot 8\cdot 7!}{2!7!}=36[/latex]
- [latex]\left(\begin{array}{c}9\\ 7\end{array}\right)=\frac{9!}{7!\left(9-7\right)!}=\frac{9\cdot 8\cdot 7!}{7!2!}=36[/latex]
[/hidden-answer]
Analysis
Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.
Try It
Find each binomial coefficient.
- [latex]\,\left(\begin{array}{c}7\\ 3\end{array}\right)\,[/latex]
- [latex]\,\left(\begin{array}{c}11\\ 4\end{array}\right)\,[/latex]
[hidden-answer a=”fs-id1165137653724″]
- 35
- 330
[/hidden-answer]
Using the Binomial Theorem
When we expand [latex]{\left(x+y\right)}^{n}[/latex] by multiplying, the result is called a binomial expansion, and it includes binomial coefficie