# Binomial Expansion

Satisfactory Essays
479 Words
Grammar
Plagiarism
Writing
Score
Binomial Expansion
BINOMIAL THEOREM

OBJECTIVES
Recognize patterns in binomial expansions.
Evaluate a binomial coefficient.
Expand a binomial raised to a power.
Find a particular term in a binomial expansion
Understand the principle of mathematical induction.
Prove statements using mathematical induction.

Definition: BINOMIAL THEOREM

Patterns in Binomial Expansions
A number of patterns, as follows, begin to appear when we write the binomial expansion of a  b n, where n is a positive integer.

a  b   a  b
a  b 2  a 2  2ab  b 2
a  b 3  a3  3a 2b  3ab 2  b3
a  b 4  a 4  4a3b  6a 2b2  4ab3  b 4
5
a  b   a5  5a 4b  10a3b 2  10a 2b3  5ab4  b5
1

and so on.

In each expanded form above, the following can be observed: n 1. The first term is a , and the exponent on a decreases by 1 in each successive term.

2. The last term is b n and the exponent on b decreases by 1 in each successive term.
3. The sum of the exponents on the variables in any term is equal to n. n n  1 terms in the expanded form of a  b .
4. There are

Definition:
Binomial Coefficients
An interesting pattern for the coefficients in the binomial expansion can be written in the following triangular arrangement n=0 n=1 n=2 n=3 n=4 n=5 n=6 a  bn

1
1
1
1

1
1

1

1
2

3

4

1
3

6

1

4

1

5
10
10
5
1
6 15
20
15
6
1

This triangular array of coefficients is called the Pascal’s Triangle.
When n is small, the use of Pascal’s triangle is advantageous.
However, if n is large or a specific term is desired, the use of Binomial Theorem is more appropriate.

Definition :
THE BINOMIAL THEOREM
The Binomial Theorem provides a formula for expanding expressions of the form a  b n , where n is a natural number.
For any binomial a  b  and any natural number n ,

n n1 nn  1 n2 2 nn  1n  2...n  r  1 nr r
a  b  a  a b  a b  ...  a b  ...  b n
1!
2!

## You May Also Find These Documents Helpful

• Powerful Essays

1. Calculate the value of “n”. Show your work. Include an example of ALL calculations involving solving for “n”.…

• 723 Words
• 3 Pages
Powerful Essays
• Satisfactory Essays

Parameters are known, constant values that are usually coefficients of variables in equations.   Answer True  False…

• 741 Words
• 3 Pages
Satisfactory Essays
• Satisfactory Essays

## hiihihi

• 10962 Words
• 41 Pages

a. Explain how long division of a polynomial expression by a binomial expression of the form…

• 10962 Words
• 41 Pages
Satisfactory Essays
• Satisfactory Essays

## heyy

• 1170 Words
• 5 Pages

Solutions: 1a 2c 3c 4d 5d 6c 7c 8d 9d 10a 11d 12a 13c 14c 15d 16c 17c 18b 19b 20c 21d 22a…

• 1170 Words
• 5 Pages
Satisfactory Essays
• Satisfactory Essays

b. Multiples b times 4 and stores the result in a //Ans.)A = B * 4…

• 453 Words
• 3 Pages
Satisfactory Essays
• Satisfactory Essays

B C _ _______ ____________ D _ _______ ____________ A _ _______ ____________ B _ _______ ____________ D _ _______ ____________ C _ _______ ____________ B _ _______ ____________ A _ _______ ____________ A _ _______ ____________ A _ _______ ____________ D _ _______ ____________…

• 1450 Words
• 6 Pages
Satisfactory Essays
• Satisfactory Essays

ASA College Department of Arts and Sciences MAT 105 College Math Final Exam Review A. Properties of Exponents: 1) −8 2 2) −3 ∙2 3) 2 3…

• 447 Words
• 2 Pages
Satisfactory Essays
• Good Essays

NB. Your assessor may wish to ask you some questions relating to this activity. There is a separate sheet for recording these. The person who observed/witnessed your activity must sign and date overleaf.…

• 3385 Words
• 15 Pages
Good Essays
• Satisfactory Essays

the ﬁber. Specify which one of the following sequences of events, {Ek , k 1} is increasing,…

• 509 Words
• 5 Pages
Satisfactory Essays
• Good Essays

To solve you use a formula for an arithmetic series because for every row, the number of cans…

• 378 Words
• 2 Pages
Good Essays
• Satisfactory Essays

Score B 1 1 2 3 4 Score A 5 6 7 8 9 10 11 12 1 1 2 3 4 6 7 8 9 10 11 12 2 1 2 3 4 4 6 7 8 9 10 11 12 3 1 2 3 4 4 6 7 8 9 10 11 12 4 2 3 3 4 5 7 8 9 10 11 11 12 5 3 4 4 5 6 8 9 10 10 11 12 12 6 3 4 5 6 7 8 9 10 10 11 12 12 7 4 5 6 7 8 9 9 10 11 11 12 12 8 5 6 7 8 8 9 10 10 11 12 12 12 9 6 6 7 8 9 10 10 10 11 12 12 12 10 7 7 8 9 9 10 11 11 12 12 12 12 11 7 7 8 9 9 10 11 11 12 12 12 12…

• 1068 Words
• 5 Pages
Satisfactory Essays
• Good Essays

MA1 MA1 A1 MA1 A1 MA1 MA2 MA1 A1, A1 A1, A1 MA1 A1 MA1 A1 8 6 6…

• 1821 Words
• 8 Pages
Good Essays
• Satisfactory Essays

To further investigate the relationship between Sn and n, they will be plotted, using technology. Relation between Sn and n (when a=2) n Sn 0 1.000000 1 1.693147 2 1.93373688 3 1.9888778 4 1.99849593 5 1.99982928 6 1.99998338 7 1.99999857 8 1.99999989 9 1.99999999 10 2.000000 A very interesting pattern can be observed from the above plot. As n increases, Sn tends to 2. Thus, as n approaches…

• 1462 Words
• 6 Pages
Satisfactory Essays
• Satisfactory Essays

a 3  2a 2 a 2  6 a  5  2 a2  4 a  3a  10…

• 1398 Words
• 6 Pages
Satisfactory Essays
• Satisfactory Essays

All A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 0 1 2 3 4 5 6 7 8 9 Other…

• 714 Words
• 3 Pages
Satisfactory Essays