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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
Practice Key Terms 3

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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