Question

explain binomial theorem​

Answer 1

A binomial is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.

We sometimes need to expand binomials as follows:

(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 + 6a2b2 + 4ab3 + b4

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.

Computers can do this for us very easily.

Pascal's Triangle

We note that the coefficients (the numbers in front of each term) follow a pattern. [This was noticed long before Pascal, by the Chinese.]

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

You can use this pattern to form the coefficients, rather than multiply everything out as we did above.

The Binomial Theorem

We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases

Binomial Theorem Formula

Based on the binomial properties, the binomial theorem states that the following binomial formula is valid for all positive integer values of n

(a + b)n = nC0an + nC1an−1b + nC2an−2b2 + nC3an−3b3 + ... + nCnbn

We can use the nCr

n

C

r

button on our calculator to find these values.

nCr

n

C

r

is sometimes written nCr.

Answer 2

Binomial theorem is a field in mathematics that deals with the expansion of a Binomial or say two degree eqn having large degree of powers.

Hope you get your answer.