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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 $\text{\hspace{0.17em}}{(x+y)}^{n}\text{\hspace{0.17em}}$ without multiplying the binomial by itself $n$ times.
In Counting Principles , we studied combinations . In the shortcut to finding $\text{\hspace{0.17em}}{(x+y)}^{n},\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ r\end{array}\right)\text{\hspace{0.17em}}$ instead of $C(n,r),$ but it can be calculated in the same way. So
The combination $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ r\end{array}\right)\text{\hspace{0.17em}}$ is called a binomial coefficient . An example of a binomial coefficient is $\text{\hspace{0.17em}}\left(\begin{array}{c}5\\ 2\end{array}\right)=C(5,2)=10.\text{\hspace{0.17em}}$
If $n$ and $r$ are integers greater than or equal to 0 with $n\ge r,$ 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.
Find each binomial coefficient.
Use the formula to calculate each binomial coefficient. You can also use the ${n}_{}{C}_{r}$ function on your calculator.
Find each binomial coefficient.
When we expand ${(x+y)}^{n}$ by multiplying, the result is called a binomial expansion , and it includes binomial coefficients. If we wanted to expand ${(x+y)}^{52},$ we might multiply $(x+y)$ by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
First, let’s examine the exponents. With each successive term, the exponent for $x$ decreases and the exponent for $y$ increases. The sum of the two exponents is $n$ for each term.
Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:
These patterns lead us to the Binomial Theorem , which can be used to expand any binomial.
Another way to see the coefficients is to examine the expansion of a binomial in general form, $\text{\hspace{0.17em}}x+y,\text{\hspace{0.17em}}$ to successive powers 1, 2, 3, and 4.
Can you guess the next expansion for the binomial $\text{\hspace{0.17em}}{(x+y)}^{5}?\text{\hspace{0.17em}}$
See [link] , which illustrates the following:
To determine the expansion on ${(x+y)}^{5},$ we see $n=5,$ thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of $x,$ the pattern is as follows:
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