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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 ( x + y ) n without multiplying the binomial by itself n times.

Identifying binomial coefficients

In Counting Principles , we studied combinations . In the shortcut to finding ( x + y ) n , 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 ( n r ) instead of C ( n , r ) , but it can be calculated in the same way. So

( n r ) = C ( n , r ) = n ! r ! ( n r ) !

The combination ( n r ) is called a binomial coefficient . An example of a binomial coefficient is ( 5 2 ) = C ( 5 , 2 ) = 10.

Binomial coefficients

If n and r are integers greater than or equal to 0 with n r , then the binomial coefficient    is

( n r ) = C ( n , r ) = n ! r ! ( n r ) !

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.

  1. ( 5 3 )
  2. ( 9 2 )
  3. ( 9 7 )

Use the formula to calculate each binomial coefficient. You can also use the n C r function on your calculator.

( n r ) = C ( n , r ) = n ! r ! ( n r ) !
  1. ( 5 3 ) = 5 ! 3 ! ( 5 3 ) ! = 5 4 3 ! 3 ! 2 ! = 10
  2. ( 9 2 ) = 9 ! 2 ! ( 9 2 ) ! = 9 8 7 ! 2 ! 7 ! = 36
  3. ( 9 7 ) = 9 ! 7 ! ( 9 7 ) ! = 9 8 7 ! 7 ! 2 ! = 36
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find each binomial coefficient.

  1. ( 7 3 )
  2. ( 11 4 )

  1. 35
  2. 330

Got questions? Get instant answers now!

Using the binomial theorem

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.

( x + y ) 2 = x 2 + 2 x y + y 2 ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4

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:

( n 0 ) , ( n 1 ) , ( n 2 ) , ... , ( n n ) .

These patterns lead us to the Binomial Theorem , which can be used to expand any binomial.

( x + y ) n = k = 0 n ( n k ) x n k y k = x n + ( n 1 ) x n 1 y + ( n 2 ) x n 2 y 2 + ... + ( n n 1 ) x y n 1 + y n

Another way to see the coefficients is to examine the expansion of a binomial in general form, x + y , to successive powers 1, 2, 3, and 4.

( x + y ) 1 = x + y ( x + y ) 2 = x 2 + 2 x y + y 2 ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4

Can you guess the next expansion for the binomial ( x + y ) 5 ?

Graph of the function f_2.

See [link] , which illustrates the following:

  • There are n + 1 terms in the expansion of ( x + y ) n .
  • The degree (or sum of the exponents) for each term is n .
  • The powers on x begin with n and decrease to 0.
  • The powers on y begin with 0 and increase to n .
  • The coefficients are symmetric.

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:

Questions & Answers

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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