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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

find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
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Nani
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0.75
Lynne
0.75
Inkoom
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duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
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Cromwell
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x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
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Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
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P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
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A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
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Sin(A+B) = sinBcosA+cosBsinA
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immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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7.5 and 37.5
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5x+x=45
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192
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Practice Key Terms 3

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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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