What is Binomial Theorem and how is it derived?

Binomial theorem is quite an interesting subject in Mathematics because of its usage in a wide range of applications. So, before going on with the utilization, let us come up with an inquiry:

What exactly is the formula for (a+b)² or (a+b)³? It may be easier for students to come up with the answer that (a+b)² is equal to a² + 2ab + b² and (a+b)³ is equal to a³ + 3a²b + 3ab² + b³. But when it comes to (a + b)⁵, it becomes quite difficult to answer. Another difficult inquiry would be, what is the answer to the equation, (a + b)⁹ or (5a – 2b)⁷? For this exact reason, the Binomial Theorem was introduced. Eventually, the equations like (a + b)² or (a – b)² also similarly use the binomial theorem formula which we will discuss later in this article.

Many students can easily equate the above using binomial theorems who are profound in mathematics and have opted for their subject during their schooling days. Even those who do not know would also get a basic idea while going through this article. Thus, what exactly is Binomial Expansion? Theoretically, the answer is quite simple. Binomial is the sum or difference between two different numbers and the expansion refers to the number of times the combined numbers are expanded.

What exactly is the Binomial Theorem?

The binomial Theorem is generally an expansion of a binomial expression to its finite or a definite power. This is majorly an algebraic expression between two distinct numbers or terms. The complete set of expansion of binomial powers using algebra is termed as Binomial theorem and it uses Pascal’s triangle to find out the coefficients.

Thus a binomial theorem for a non-negative integer n can be expressed as:

(a+b)n= n ∑r=0 (nr)an–rbr

where (nr)=nCr=n!/r!(n–r)! is the binomial coefficient.

So, how does a Pascal’s Triangle help in determining the coefficient of a Binomial Expression?

(a + b)⁰ = 1

(a + b)¹ = a +b

(a + b)² = a² + 2ab +b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a + b)⁴ = a⁴ + 4a³b + 4²b² + 4ab³ + b⁴

(a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + b⁵

So, these expressions display many patterns, such as:

  • Each expansion consists of one extra term as regarded with the power of the binomial.
  • The power of the binomial is the same as the sum of the exponents, found in each term within the expansion.
  • The power of ‘a’ is reduced by one with each particular term whereas the power of ‘b’ increases by one with each successive term.
  • The coefficients form a typical pattern that is symmetrical.

Thus the coefficients of the Binomial Theorem are identified and are identical to the entries within the nth row found in Pascal’s triangle.

The expansion of (a+b)n can be expressed as:

(a + b)n = nC0anb0 + nC1an-1b1 + nC2an-2b2 + ….. + nCna0bn

(a+b)n= n∑r=0 (nr)an–rbr where nCr = n!/r!(n–r)!

Important Points to remember

  • If (a + b)n is expanded, the total number of terms tends to n+1.
  • The addition of the powers of a and b is always n.
  • nC0, nC1, nC2 , ………. nC3 are all binomial coefficients.

To know about the various fundamentals, do visit Cuemath for their math online classes. You can learn mathematics and coding using various tips and tricks.

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