<< Chapter < Page Chapter >> Page >

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 ) ! .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 3 a + 2 b ) 3

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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( a 2 b ) 15

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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

( x 3 y ) 8

x 24 8 x 21 y + 28 x 18 y

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

The fourth term of ( 3 x 2 y ) 5

720 x 2 y 3

Got questions? Get instant answers now!

The third term of ( 6 x 3 y ) 7

Got questions? Get instant answers now!

The eighth term of ( 7 + 5 y ) 14

220 , 812 , 466 , 875 , 000 y 7

Got questions? Get instant answers now!

The seventh term of ( a + b ) 11

Got questions? Get instant answers now!

The fifth term of ( x y ) 7

35 x 3 y 4

Got questions? Get instant answers now!

The tenth term of ( x 1 ) 12

Got questions? Get instant answers now!

The ninth term of ( a 3 b 2 ) 11

1 , 082 , 565 a 3 b 16

Got questions? Get instant answers now!

The fourth term of ( x 3 1 2 ) 10

Got questions? Get instant answers now!

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

1152 y 2 x 7

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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

k 1

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Questions & Answers

the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask