1. What is Binomial Expression?
Any expression that has two terms is called a binomial expression.
Eg: x + y, x + 2, 2x + y, 3x + 4y, 3x² + 4y³.
Now,let us make the following observations based on the above five algebraic formulas:
1. The coefficients in each of the above binomial expansions follow a pattern. The pattern can be understood with Pascal’s triangle as follows:
2. First let us become familiar with the values of the binomial coefficients
(see point no. 4 in red below for a detailed discussion of binomial coefficients)
Remember the Combinations Formula ncr ?
and use it to write the following sample values in the binomial expansion no.5 above:
3. The first term in each of the above binomial expansions is xn or nC0 xn
The second term in the above expansions is nxn-1
4. As the expansion proceeds the power of x decreases by one, while the power of y increases by one.
5. The number of terms in each of the expansion is (n+1)
6. Note: the last term is (n + 1) th term, not nth term.
7. Generalizing the above properties in the above four algebraic expansions, we can write the following general binomial theorem:
3. Number of terms in the binomial expansion: (x + y)n
1. The number of terms in the binomial expansion is always one more than the index n,
i.e., in a binomial expansion, number of terms = n + 1, where n is the index.2. As the terms are written, the power of x decreases by one, while the power of y increases by one.