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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

By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
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Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
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Miranda
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Miranda
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Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
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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