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The binomial theorem (or binomial expansion) is a consequence of increasing the powers of binomials or sums of two terms.

The coefficients of the terms in the extension are the binomial coefficients.

The theorem and its simplification can be used to establish results and resolve problems in combinatorics, algebra, calculus, and several more areas of mathematics.

A formula in which each positive integral power of a known binomial expression or equation can be developed or expanded in a matter of series which are termed as the binomial theorem.

We already know how to discover the squares and cubes of binomials like a + b and a – b.

E.g. (a+b)2 , (a-b)3  etc.

However, for higher powers calculation and computation become complicated. We are able to rise above this obstacle by using the binomial theorem.

It provides for us an easier way to expand for us (a + b)n, where n is an integer or a rational number.

Image Credits: Exam Fear

The Interesting History of the Binomial Theorem

Unique instances of the binomial theorem were known since the 4th century B.C. when Greek mathematician Euclid mentioned the extraordinary case of the binomial theorem for exponent 2.

There is proof that the binomial theorem for cubes was known by the 6th century in India.

The binomial theorem as such can be found in the work of 11th-century Persian mathematician Al-Karaji, who portrayed the triangular pattern of the binomial coefficients.

He also presented mathematical evidence of both the binomial theorem and Pascal’s triangle, using an early form of mathematical induction.

The Persian poet and mathematician Omar Khayyam was possibly recognizable with the formula for higher orders, even though, unfortunately, many of his mathematical works are missing.

Properties of the Binomial Theorem

  • The total number of each and every term in the expansion of the theorem is (x+a)n is (n+1)
  • The sum total of the indices of x and a in each term is n
  • The expansion shown above is also true when both x and a are complex numbers.
  • The coefficient of all the terms is equidistant (equal in distance from each other) from the beginning to the end. The coefficients are termed as the binomial coefficients.
  • The values of these binomial coefficients gradually go up to the maximum and the progressively lessen.

Middle Term in the Expansion of (1+x)n

  • If the value of n is an even number then, in the elongation or expansion of (x+a)n, the middle term is (n/2+1)th
  • In the case of the opposite scenario, where the value of n is an odd number, the elongation or expansion of the middle term will be (n+1)/2th term and (n+3)/2th term.

Greatest Coefficients

  • In the case that the value of n is even, then for the case of (x+a)n, the highest or greatest co-efficient will be nCn/2.
  • In the case of the opposite scenario, where the value of n is odd, then in the term (x+a)n, the highest or greatest co-efficient will be nCn-1/2 or nCn-1/2 the two being exactly the same or equal.

Greatest Terms                                         

Consider the expansion or elongation of (x+a)n,

  • If n+1/ x/a + 1 is an integer = p, then the greatest term will be Tp = = Tp +1
  • In the case of the opposite scenario, where n+1/ x/a is not an integer, along with m being a vital part of n+1 /x/a+1, then Tm+1 will be the greatest term.

Important points to remember while studying the Binomial Theorem

  • The full amount of the 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 declining by 1 while the powers of the second quantity ‘b’ augment by 1, in the consecutive terms.
  • In each term of the expansion, the sum total of the indices of a and b is the same and is equal to the index of a + b.
  • If in the case that n is a positive integer, then (1+x)n will have (n+1) terms which means that there will be only a specific and finite number of terms. In the case that n is a regular or general exponent, then the elongation or expansion of (1+x)n will for sure contain an infinite number of terms.
  • In the same scenario where n is a positive integer, the expansion or elongation of (1+x)n will be applicable for each and every value of x. similarly, in the same scenario as mentioned in the previous point, if n is the general exponent, then in that case, the expansion or elongation of (1+x)n  will be suitable for the values of x satisfying the condition of x < 1.

The following is a numerical question based on the rule of the binomial theorem as mention in the first few paragraphs. Try and calculate the answer to check that you’ve understood the concept of the lesson.

Compute (98)5

Solution:  (98)5 = (100-2)5   =

= 5C0 (100)5 – 5C1 (100)4.2 + 5C2 (100)322 – 5C3 (100)2 (2)3 + 5C4 (100) (2)4 – 5C5 (2)5

= 10000000000 – 5 × 100000000 × 2 + 10 × 1000000 × 4 – 10 ×10000 × 8 + 5 × 100 × 16 – 32

= 10040008000 – 1000800032 = 9039207968

This was a brief introduction to Binomial Theorem. To understand the application of derivatives, click here.

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