# Use the binomial theorem to expand the binomial (s+2v)^5.

## Question:

Use the binomial theorem to expand the binomial {eq}(s+2v)^5\ {/eq}.

## Expansion using Binomial theorem

As per the Binomial theorem, we can easily find the expansion of binomial expressions raised to certain power. This would be a cumbersome and error-prone exercise if we use algebraic multiplication for expansion.

The theorem states that an algebraic expression {eq}(p + q)^n = \sum_{k = 0}^{n}{\binom{n}{k}}(p^{n-k})(q^k) {/eq}

where, {eq}\binom{n}{k} = \ \frac{n!}{k!(n-k)!} {/eq}

The binomial expansion of {eq}(s+2v)^5 = \sum_{k = 0}^{5}{\binom{5}{k}}(s^{5-k})(2v)^k {/eq}

=> {eq}(s+2v)^5 = s^5 + 5s^4(2v) + 10s^3(4v^2) + 10s^2(8v^3) + 5s(16v^4) + 32v^5 {/eq}

=> {eq}(s+2v)^5 = s^5 + 10s^4v + 40s^3v^2 + 80s^2v^3 + 80sv^4 + 32v^5 {/eq}