Binomial Theorem

Binomial Expressions

As the power increases the expansion of terms becomes very lengthy and tedious to calculate. It can be easily calculated with the help of the Binomial Theorem. 


What is Binomial Theorem?

The binomial theorem in mathematics is the process of expanding an expression that has been raised to any finite power. A binomial theorem is a powerful tool of expansion, which is widely used in Algebra, probability, etc.


Binomial Expression 

A binomial expression is an algebraic expression that contains two dissimilar terms such as a + b, a³ + b³, etc.


Binomial Theorem Expansion

According to the theorem, we can expand the power (x + y)\[^{n}\] into a sum involving terms of the form ax\[^{b}\]y\[^{c}\], where the exponents b and c are nonnegative integers with b+c=n and the coefficient a of each term is a specific positive integer depending on n and b.

The theorem is given by the formula:

(x + y)\[^{n}\] = \[\sum_{k=0}^{n}\] (\[_{k}^{n}\]) x\[^{n-k}\]y\[^{k}\] = \[\sum_{k=0}^{n}\] (\[_{k}^{n}\]) x\[^{k}\]y\[^{n-k}\]

(x + y)\[^{n}\] = \[\sum_{k=0}^{n}\] (nk) x\[^{n-k}\]y\[^{k}\] = \[\sum_{k=0}^{n}\] (nk) x\[^{k}\]y\[^{n-k}\]


Binomial Theorem Rules

The coefficients that appear in the binomial expansion are known as binomial coefficients. These are usually written (\[_{k}^{n}\]) or \[ ^{n}C_{k}\]. which means n choose k.

The coefficient of a term x\[^{n-k}\]y\[^{k}\] in a binomial expansion can be calculated using the combination formula. The formula consists of factorials:

(\[_{k}^{n}\]) = \[\frac{n!}{k!(n-k)!}\] 


Important Points to Remember While Solving Binomial Expansion:

  • The total number of terms in the expansion of (x + y)\[^{n}\] is (n+1)

  • The sum of exponents is always equal to n i.e (x + y) = n.

  • nC\[_{0}\], nC\[_{1}\], nC\[_{2}\], … .., nC\[_{n}\] is called binomial coefficients and also represented by C\[_{0}\], CC\[_{1}\], CC\[_{2}\] ….., C\[_{n}\] respectively.

  • The binomial coefficients which are equidistant from the beginning and from the ending are of equal value i.e. nC0= nCn,nC1= nCn-1 , nC2= nCn-2 ,….. etc.

  • To find binomial coefficients we can also use Pascal’s Triangle.


Some Other Useful Expansions that Help in an Easy Way to Solve Binomial Theorem :

  • (x + y)\[^{n}\] + (x - y)\[^{n}\] = 2[C\[_{0}\]x\[^{n}\] + C\[_{2}\]x\[^{(n-1)}\]y\[^{2}\] + C\[_{4}\]x\[^{n-4}\]y\[^{4}\]+ …]

  • (x + y)\[^{n}\] - (x - y)\[^{n}\] = 2[C\[_{1}\] x\[^{(n-1)}\]y + C\[_{3}\] x\[^{(n-3)}\]y\[^{3}\] + C\[_{5}\]x\[^{(n-5)}\]y\[^{5}\]...]

  • (1 + x)\[^{n}\] = [C\[_{0}\] + C\[_{1}\]x + C\[_{2}\]x\[^{2}\] + … C\[_{n}\]x\[_{n}\]]

  • (1 + x)\[^{n}\] + (1 - x)\[^{n}\] = 2[C\[_{0}\] + C\[_{2}\]x\[^{2}\] + C\[_{4}\]x\[^{4}\] + …]

  • (1 + x)\[^{n}\] - (1 - x)\[^{n}\] = 2[C\[_{1}\]x + C\[_{3}\]x\[^{3}\] + C\[_{5}\]x\[^{5}\] + …]

  • The number of terms in the expansion of (x + a)\[^{n}\] + (x - a)\[^{n}\] is (n+2)/2 if “n” is even or (n+1)/2 if “n” is odd.

  • The number of terms in the expansion of (x + a)\[^{n}\] - (x - a)\[^{n}\] is (n/2) if “n” is even or (n+1)/2 if “n” is odd.

Properties of Binomial Coefficients

Binomial coefficients refer to the integers that are coefficients in the binomial theorem. Some of the important properties of binomial coefficients are given below:

  • C\[_{0}\] + C\[_{1}\] + C\[_{2}\] + … + C\[_{n}\] = 2n

  • C\[_{0}\] + C\[_{2}\] + C\[_{4}\] + … = C\[_{1}\] + C\[_{3}\] + C\[_{5}\]... = 2n - 1

  • C\[_{0}\] - C\[_{1}\] + C\[_{2}\] - C\[_{3}\] + … + (-1)\[^{n}\] . nC\[_{n}\] = 0

  • nC\[_{1}\] + 2nC\[_{2}\] + 3.nC\[_{3}\] + … +n.nC\[_{n}\] = n.2\[^{(n-1)}\].

  • C\[_{1}\] - 2C\[_{2}\] + 3C\[_{3}\] - 4C\[_{4}\]+ … +(-1)\[^{(n-1)}\]C\[_{n}\] = 0 for n > 1

  • C\[_{0}\] \[^{2}\] + C\[_{1}\] \[^{2}\] + C\[_{2}\] \[^{2}\] + ...C\[_{n}\] \[^{2}\] = \[\frac{[(2n)!]}{(n!)^{2}}\]

Terms in the Binomial Expansion

In binomial expansion, we generally find the middle term or the general term. The different Binomial Term involved in the binomial expansion is:

  • General Term

  • Middle Term

  • Independent Term

  • Determining a Particular Term

  • Numerically greatest term

  • The ratio of Consecutive Terms/Coefficients

General Term in Binomial Expansion:

We have (x + y)\[^{n}\] = nC\[_{0}\]x\[^{n}\] + nC\[_{1}\]x\[^{(n-1)}\]y + nC\[_{2}\]x\[^{n-2}\]y\[^{2}\] + … + nC\[_{n}\]Y\[^{n}\]

General Term = T\[_{(r+1)}\] = nC\[_{r}\]x\[^{n-r}\] .y\[^{r}\]

General Term in (1+x)\[^{n}\] nC\[_{r}\]x\[^{r}\]

In the binomial expansion of (x + y)\[^{n}\], the r\[^{th}\] term from the end is (n - r + 2)\[^{th}\].

Middle Term(S) in the expansion of (x + y)\[^{n,n}\] 

If n is even then (n/2 + 1) term is the middle term.

If n is odd then [(n+1)/2]\[^{th}\] and [(n+3)/2)\[^{th}\] terms are the middle terms of the expansion.


Applications of Binomial Theorem

The binomial theorem has various applications in mathematics like finding the remainder, finding digits of a number, etc. The most common binomial theorem applications are:

  • Finding Remainder using Binomial Theorem.

  • Finding Digits of a Number.

  • Relation Between two Numbers.

  • Divisibility Test.

Binomial Theorem Problems are explained with the help of Binomial theorem formula examples which is given below:

1. Find the coefficient of x\[^{9}\] in the expansion of (1 + x) (1 + x\[^{2}\]) (1 + x\[^{3}\]) . . . . . . (1 + x\[^{100}\]).

Sol: x\[^{9}\] can be formed in 8 ways.

i.e., x\[^{9}\] x\[^{(1+8)}\] x\[^{(2+7)}\] x\[^{(3+6)}\] x\[^{(4+5)}\], x\[^{(1+3+5)}\], x\[^{(2+3+4)}\]

∴ The coefficient of x\[^{9}\] = 1 + 1 + 1 + . . . . + 8 times = 8.

2.Find the number of terms in (1 + 2x + x\[^{2}\])\[^{50}\].

Sol: Use formula of the number of terms (1 + 2x + x\[^{2}\])\[^{50}\] = [(1 + x)\[^{2}\]]\[^{50}\] = (1 + x)\[^{100}\]

Hence, the number of terms = (100 + 1) = 101

3. Find the value of $\binom{6}{2}$ using a pascal triangle.

Solution:

Look at the 2nd element in the 6th row in pascal's triangle. The value of $\binom{6}{2}$ will be that element.

Hence, the value of $\binom{6}{2}$ is $15$.


4. Expand $( a + 2)^6$ using binomial theorem.

Solution:

Let $a = x, y = 2$ and $n = 6$

Substituting the values on binomial formula, we get

$(a)^6+6(a)^5(2)+\dfrac{6(5)}{2\text{!}}(a)^4(2)^{2}+\dfrac{6(5)(4)}{3\text{!}}(a)^3(2)^3+\dfrac{6(5)(4)(3)}{4\text{!}}(a)^2(2)^4+\dfrac{6(5)(4)(3)(2)}{5\text{!}}(a)(2)^5+ 2^6$

$= a^6 + 12a^5 + 60a^4 +160a^3 + 240a^2 + 192a + 64$


Facts to Remember

  • The binomial theorem was invented by Issac Newton.

  • The Pascal triangle was invented by Blaise Pascal.

  • The numbers in each row in the pascal triangle are known as the binomial coefficients.

  • The numbers on the second diagonal and third diagonal in the pascal triangle form counting numbers and triangular numbers respectively.

  • The sum of the numbers on each row are powers of $2$ whereas a series of diagonals of pascal's triangle forms the Fibonacci Sequence.

FAQs (Frequently Asked Questions)

1.How Do You Use the Binomial Theorem?

Ans: The Binomial theorem help us how to expand expressions of the form (a+b)ⁿ, for example, (x+y)⁷. The larger the power is, it becomes very difficult to expand expressions like this directly. But with the Binomial theorem, we can expand relatively fast.

2.How Do You Expand a Binomial Expression Using Pascal's Triangle?

Ans: In Pascal’s triangle every row is built from the row above it. It gives us the coefficients for an expanded binomial of the form (a + b)n, where n is the row of the triangle. We can use these coefficients to find the entire expanded binomial, with a couple of extra tricks