Finding Remainder Using Binomial Theorem

A binomial expression is an algebra that consists of two terms connected by either a positive or a negative sign, i.e. they are in either the addition relation or the subtraction relation. For example, a + 2y and a – 2y, are binomial algebraic terms. The Binomial Theorem stated by the famous Greek mathematician Euclid in the 4th century helps us in finding the expanded values of the algebraic expression that are in the form of  (x + y)n. The theorem is most important in calculating the terms with a greater value for ‘n’. For example, any algebraic expression such as (x + y)2 and  (x + y)3 can be calculated just with basic algebraic treatments. Whereas the calculation of the expanded form of the terms with greater exponents such as (x + y)23 becomes quite difficult. To eradicate this problem, the Binomial Theorem has proved to be of great use. The value of the exponent can even be either negative or fraction while calculating via this theorem. Let’s study the article to get the Binomial Theorem JEE notes.

Binomial Theorem Formula

As stated above, the Binomial Formula helps us in determining the expansion of any power of the binomial, i.e. in the form of a series.

Thus, the formula for the Binomial Theorem is as follows:

(a +b)n = ∑nr=0(n C r)an−r br

a and b = real numbers,

n = positive integer

C = coefficient

Binomial Coefficient: The coefficient of the binomial is the number concerning the variables of the binomial under expansion. We can use the Pascal Triangle or the Combinations Formula to obtain the Binomial Coefficient represented as:

nC0, nC1, nC2, nC3, nC4…

Binomial Expansion of the Negative Exponents

As stated above, we can also expand the binomials with negative exponents using the Binomial Theorem. For the negative exponents, the coefficient values and the terms of the expansion remain the same; however, the algebraic relationship between the terms varies from the binomials with the positive exponents.

Features of Binomial Theorem

The basic features of the Binomial Theorem are as follows:

·  For a given binomial expansion as (x + y)n, the number of coefficients will be equal to (n+1).

·  For the given expression (x + y)n,  the terms will be (n+1). Moreover, the first term will be xn and the last term will be yn.

·  During the expansion, the power of x will decrease from the original value (say n) up to 0.

·  In contrast, the value of a will increase from 0 to the nth value.

Pascal’s Triangle

An alternative way of determining the coefficients of the binomial expansion is Pascal’s Triangle. It got its name from the famous French mathematician Blaise Pascal. In this method, we use the diagram instead of the algebraic calculation for determining the coefficients. The coefficients of the binomial expression are in the special trend; therefore, it is possible to observe them in the form of a diagram, i.e. Pascal’s triangle diagram. The arrangement of the numbers within this triangle has 1 as their border element. For the other remaining numbers, the arrangement is in such a way that the numbers are the sum of the other two numbers, i.e. above them. It is an effective way of determining the coefficients for the binomials having small exponents.

Binomial Theorem Questions

Q1. Find the remainder using the binomial theorem if 5103 gets divided by 13.

Solution: 

Expressing the in 5103 terms of 13, we get

= 5×5102

= 5×52×51

= 5×2551

= 5×(26−1)51

= 5×(2×13−1)51

Using the Binomial Formula, we get:

= ∑51r=0(-1)r 51Cr2651-r

= 5× (2×13−1)51≡5×51C51 (−1)51

= 5× (2×13−1)51≡−5

= 5× (2×13−1)51≡−5+13≡8

Answer: As we know, the remainder can never be zero; therefore, the value for the remainder using the binomial theorem if 5103 gets divided by 13 shall be 8.

Q2. Expand the binomial term given as (x+3)5 using the Binomial Theorem. The value of a=3 and n=5.

Solution:

According to the questions,

Binomial Expression = (x+3)5

a = 3 and

n =5

On expansion using the theorem of binomial, we get:

= (x+3)5 = 5C0x5−0 30+5C1x5−1 31+5C2x5−2 32+5C3x5−3 33+5C4x5−4 34+5C5x5−5 35

 x5 + 5 x4. 3 + 10 x3 . 9 + 10 x2 . 27 + 5x .81 + 35

Answer: The binomial term given as (x+3)5 using the binomial theorem will be x5 + 15 x4 + 90x3 + 270 x2 + 405 x + 243

Q3. What shall be the 7th term in the (x + 2)10 binomial expression?

Solution:

For the calculation of any general term, we can use the formula as follows:

Tr+1 = nCr xn−ryr

According to the question,

r = 6

n = 10, and

a = 2

Now, we must substitute the values given in the question in the formula:

T6+1 = 10C6x10−626

T7 = 210 x 4. 64

= 13440 x4

Answer: The 7th term in the (x + 2)10 binomial expression is 13440 x4

Conclusion

As the name suggests, the binomial term comprises two terms separated by a + or a – sign in between. The Binomial Theorem stated by famous Greek Mathematician Euclid in the 4th century helps us in finding the expanded values of the algebraic expression that are in the form of (x + y)n. The formula of the Binomial Theorem is  (a +b)n = ∑nr=0(n C r)an−r br. As stated above, for a given binomial expansion as (x + y)n, the number of coefficients will be equal to (n+1).