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Key equations

Binomial Theorem ( x + y ) n = k 0 n ( n k ) x n k y k
( r + 1 ) t h term of a binomial expansion ( n r ) x n r y r

Key concepts

  • ( n r ) is called a binomial coefficient and is equal to C ( n , r ) . See [link] .
  • The Binomial Theorem allows us to expand binomials without multiplying. See [link] .
  • We can find a given term of a binomial expansion without fully expanding the binomial. See [link] .

Section exercises

Verbal

What is a binomial coefficient, and how it is calculated?

A binomial coefficient is an alternative way of denoting the combination C ( n , r ). It is defined as ( n r ) = C ( n , r ) = n ! r ! ( n r ) ! .

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What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

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What is the Binomial Theorem and what is its use?

The Binomial Theorem is defined as ( x + y ) n = k = 0 n ( n k ) x n k y k and can be used to expand any binomial.

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When is it an advantage to use the Binomial Theorem? Explain.

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Algebraic

For the following exercises, evaluate the binomial coefficient.

For the following exercises, use the Binomial Theorem to expand each binomial.

( 4 a b ) 3

64 a 3 48 a 2 b + 12 a b 2 b 3

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( 3 a + 2 b ) 3

27 a 3 + 54 a 2 b + 36 a b 2 + 8 b 3

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( 4 x + 2 y ) 5

1024 x 5 + 2560 x 4 y + 2560 x 3 y 2 + 1280 x 2 y 3 + 320 x y 4 + 32 y 5

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( 4 x 3 y ) 5

1024 x 5 3840 x 4 y + 5760 x 3 y 2 4320 x 2 y 3 + 1620 x y 4 243 y 5

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( x 1 + 2 y 1 ) 4

1 x 4 + 8 x 3 y + 24 x 2 y 2 + 32 x y 3 + 16 y 4

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For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

( a + b ) 17

a 17 + 17 a 16 b + 136 a 15 b 2

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( a 2 b ) 15

a 15 30 a 14 b + 420 a 13 b 2

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( 3 a + b ) 20

3 , 486 , 784 , 401 a 20 + 23 , 245 , 229 , 340 a 19 b + 73 , 609 , 892 , 910 a 18 b 2

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( x 3 y ) 8

x 24 8 x 21 y + 28 x 18 y

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For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The fourth term of ( 2 x 3 y ) 4

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The fourth term of ( 3 x 2 y ) 5

720 x 2 y 3

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The third term of ( 6 x 3 y ) 7

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The eighth term of ( 7 + 5 y ) 14

220 , 812 , 466 , 875 , 000 y 7

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The seventh term of ( a + b ) 11

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The fifth term of ( x y ) 7

35 x 3 y 4

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The tenth term of ( x 1 ) 12

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The ninth term of ( a 3 b 2 ) 11

1 , 082 , 565 a 3 b 16

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The fourth term of ( x 3 1 2 ) 10

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The eighth term of ( y 2 + 2 x ) 9

1152 y 2 x 7

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Graphical

For the following exercises, use the Binomial Theorem to expand the binomial f ( x ) = ( x + 3 ) 4 . Then find and graph each indicated sum on one set of axes.

Find and graph f 1 ( x ) , such that f 1 ( x ) is the first term of the expansion.

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Find and graph f 2 ( x ) , such that f 2 ( x ) is the sum of the first two terms of the expansion.

f 2 ( x ) = x 4 + 12 x 3

Graph of the function f_2.
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Find and graph f 3 ( x ) , such that f 3 ( x ) is the sum of the first three terms of the expansion.

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Find and graph f 4 ( x ) , such that f 4 ( x ) is the sum of the first four terms of the expansion.

f 4 ( x ) = x 4 + 12 x 3 + 54 x 2 + 108 x

Graph of the function f_4.
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Find and graph f 5 ( x ) , such that f 5 ( x ) is the sum of the first five terms of the expansion.

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Extensions

In the expansion of ( 5 x + 3 y ) n , each term has the form ( n k ) a n k b k ,   where   k successively takes on the value 0 , 1 , 2 , ... , n . If ( n k ) = ( 7 2 ) , what is the corresponding term?

590 , 625 x 5 y 2

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In the expansion of ( a + b ) n , the coefficient of a n k b k is the same as the coefficient of which other term?

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Consider the expansion of ( x + b ) 40 . What is the exponent of b in the k th term?

k 1

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Find ( n k 1 ) + ( n k ) and write the answer as a binomial coefficient in the form ( n k ) . Prove it. Hint: Use the fact that, for any integer p , such that p 1 , p ! = p ( p 1 ) ! .

( n k 1 ) + ( n k ) = ( n + 1 k ) ; Proof:

( n k 1 ) + ( n k ) = n ! k ! ( n k ) ! + n ! ( k 1 ) ! ( n ( k 1 ) ) ! = n ! k ! ( n k ) ! + n ! ( k 1 ) ! ( n k + 1 ) ! = ( n k + 1 ) n ! ( n k + 1 ) k ! ( n k ) ! + k n ! k ( k 1 ) ! ( n k + 1 ) ! = ( n k + 1 ) n ! + k n ! k ! ( n k + 1 ) ! = ( n + 1 ) n ! k ! ( ( n + 1 ) k ) ! = ( n + 1 ) ! k ! ( ( n + 1 ) k ) ! = ( n + 1 k )

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Which expression cannot be expanded using the Binomial Theorem? Explain.

  • ( x 2 2 x + 1 )
  • ( a + 4 a 5 ) 8
  • ( x 3 + 2 y 2 z ) 5
  • ( 3 x 2 2 y 3 ) 12

The expression ( x 3 + 2 y 2 z ) 5 cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

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Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
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
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
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
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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