Binomial Theorem for JEE
The binomial theorem helps us expand expressions like ((p + q)^n). For a positive integer (n), the expansion is:
(p + q)^n = \sum_{k=0}^{n} ^{n}\mathrm{C}_{k} p^{n-k} q^kWhere the binomial coefficient is given by:
$$ ^{n}\mathrm{C}_{k} = \dfrac{n!}{k!(n-k)!} $$
For example, let’s try ((x + \dfrac{1}{x})^2):
(x + \dfrac{1}{x})^2 = ^{2}\mathrm{C}<em>{0} x^2 \cdot \left(\dfrac{1}{x}\right)^0 + ^{2}\mathrm{C}</em>{1} x^1 \cdot \left(\dfrac{1}{x}\right)^1 + ^{2}\mathrm{C}_{2} x^0 \cdot \left(\dfrac{1}{x}\right)^2This simplifies to (x^2 + 2 + \dfrac{1}{x^2}).
