Binominal Expansion

Binominal Expansion

Concept and Rule

• Binomial expansion is a way of expanding expressions that are raised to integer powers.
• If (a + b)^n is the binomial, then the binomial theorem states that the expansion is a sum involving terms of the form a^k * b^(n−k) * C(n, k), k is an integer from 0 to n, and C(n, k) is a binomial coefficient.

Binomial Coefficient

• The binomial coefficients are the numbers which give the number of ways of choosing items from a larger set.
• They can be represented as C(n, k) = n! / [k!(n-k)!].
• These are the coefficients in the binomial expansion and also entries in Pascal’s triangle.

Pascal’s Triangle

• Pascal’s triangle is an array of the binomial coefficients.
• It starts with a single ‘1’ at the top. Each following row is formed by adding the number directly above it with the number to the left of the one above.

Application of Binomial Expansion

• Binomial expansion can be used to expand binomials involving both numerical and algebraic terms raised to a certain power.
• It can be used to approximate large powers and roots.
• It also helps to simplify calculations by reducing complex polynomial expressions to manageable or solvable forms.

Binomial theorem for Negative and Rational Indices

• Binomial theorem can be extended to expressions of the form (a + b)^r, where r is any rational number, but the series becomes an infinite series in this case.
• The general term in this expansion is T(r, k) = [(r)(r-1)(r-2)…(r-k+1)] / [k!] * a^(r-k) * b^k.
• It should be noted that for a negative or rational index, the formula results in an infinite series expansion. Therefore, convergence of the series should be checked.