Binomial Theorem - Formula, Expansion, Problems and Applications - Practice Questions & MCQ

An algebraic expression consisting of only two terms is called a Binomial Expression

If we wanted to expand (x + y)⁵², we might multiply (x + y) by itself fifty-two times. This could take hours!

But if we examine some simple expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.

On examining the exponents, we find that with each successive term, the exponent for x decreases by 1 and the exponent for y increases by 1. The sum of the two exponents is n for each term.

Also the coefficients for (x+y)ⁿ are equal to 

where, 

These patterns lead us to the Binomial Theorem, which can be used to expand any binomial expression.

For any natural number n, binomial expansion is

.

The combination  is called a binomial coefficient. 

Note: n can be any rational number for binomial expansion, but formula for n that are not natural numbers is different and will be covered in later sections.

Properties of Binomial Theorem and Binomial Coefficient

1. If n is a natural number, then the expansion (x + y)ⁿ has n + 1 terms.

2. In each term, the sum of the index of ‘x’ and ‘y’ is equal to ‘n’.

3. The sum of two consecutive binomial coefficients, 

4. 

Properties of Binomial Theorem and Binomial Coefficient

        5. Binomial coefficients of the term equidistant from the beginning and end are equal.

            Eg, second binomial coefficient from start = ⁿC₁, and second binomial coeffcient from the end = ⁿC_n-1

               These are equal as 1 + (n-1) = n

        6. 

        7.