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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
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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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