Binomial Theorem

Binomial Theorem

In this article, find out all you need to know about binomial theorem.

A binomial is an expression with two terms connected by + or – sign.

Let’s look into the following example to understand the difference between monomial, binomial and trinomial.

  • 3xy2 (Monomial term)
  • 5x – 1, 2y + 7 (Binomial term)
  • 3x + 5y2 – 3 (Trinomial term)

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form a xbyc, where the exponents b and c are non-negative 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 useful in algebra as well as for determining permutations and combinations, and probabilities. For positive integer exponents, n, the theorem was known to Islamic and Chinese mathematicians of the late medieval period.

The most basic example of the binomial theorem is the formula for the square of x + y

(x + y)2 = x2 + 2xy + y2

Understanding the binomial theorem with the help of Pascal’s Triangle

Isaac Newton wrote a generalized form of the Binomial Theorem. However, for quite some time Pascal’s Triangle had been well known as a way to expand binomials. The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the second row of Pascal’s triangle. (Note that the top “1” of the triangle is considered to be row 0, by convention.) The coefficients of higher powers of x + y correspond to lower rows of the triangle:

Let’s look at all the results we get, from (a+b)0 up to (a+b)3:

And now look at just the coefficients (with a “1” where a coefficient wasn’t shown):

They actually make Pascal’s Triangle

Each number is just the two numbers above it added together (except for the edges, which are all “1”)

The coefficients of higher powers of x + y correspond to lower rows of the triangle:

Several patterns can be observed from these examples. In general, for the expansion (x + y)n:

  1. the powers of start at n and decrease by 1 in each term until they reach 0 (with x0 = 1, often unwritten);
  2. the powers of start at 0 and increase by 1 until they reach n;

How do we write a formula for “find the coefficient from Pascal’s Triangle”?

Well, there is such a formula:

It is commonly called “n choose k” because it is how many ways to choose k elements from a set of n.

The “!” means “factorial”, for example 4! = 4×3×2×1 = 24

 

Now it can all go into one formula:

Points to note in Binomial Theorem

  • Total number of terms in expansion = index count +1.  g. expansion of (a + b)2, has 3 terms.
  • Powers of the first quantity ‘a’ go on decreasing by 1 whereas the powers of the second quantity ‘b’ increase by 1, in the successive terms.
  • In each term of the expansion, the sum of the indices of a and b is the same and is equal to the index of a + b.

General Term

  • General term in the expansion of (a+b)n is Tr+1 nCr  an–r br
  • Numerical: Find the 4th term in the expansion of (x – 2y)12
  • Solution:  Putting r = 3, n = 12, a = x & b = -2y in this formula the formula, Tr+1 nCr  an–r br
  • T412C3 (x)9(-2y)3
  • Or T4= -1760 x9 y3

 Middle Term

Consider the expansion of (a+b)n

  • Case 1: “n” is even, total term is expansion: n+1 à Odd
  • Middle term = (n/2 +1)th
  • We Can now find the middle term using the general term formula Tr+1 nCr an–r br
  • Case 2: “n” is odd, total term in expansion: n+1 à Even
  • There will be two Middle terms = ((n+1)/2)th    &  ((n+1)/2  + 1)th terms
  • We can now find the middle term using the general term formula Tr+1 nCr an–r br

Greatest Coefficient

  • If n is even, then in (x + a)n , the greatest coefficient is nCn / 2
  • If n is odd, then in (x + a)n, the greatest coefficient is nCn – 1 / 2 or nCn + 1 / 2 both being equal.

Greatest Term

In the expansion of (x + a)n

  • If n + 1 / x/a + 1 is an integer = p (say), then greatest term is Tp == Tp + 1.
  • If n + 1 / x/a + 1 is not an integer with m as integral part of n + 1 / x/a + 1, then Tm + 1. is the greatest term.

 Solved Examples

 

Important Points to be Remembered

  1. If n is a positive integer, then (1 + x)n contains (n +1) terms i.e., a finite number of terms.
  2. When n is general exponent, then the expansion of (1 + x)n contains infinitely many terms.
  3. When n is a positive integer, the expansion of (l + x)n is valid for all values of x. If n is general exponent, the expansion of (i + x)n is valid for the values of x satisfying the condition |x| < 1

Practice Questions

  1. Find the coefficient of x5 in the product (1+2x)6 (1 – x)7 using binomial theorem.
  2. Using binomial theorem, evaluate (105)5
  3. Find a b and n in the expansion (a + b)n, if the first three terms of the expansion are 729, 7290 and 30375, respectively.

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