# Find the binomial series for the function (1 + 2 x)^4.

## Question:

Find the binomial series for the function {eq}(1 + 2 x)^4 {/eq}.

## Binomial Theorem

{eq}{/eq}

The Binomial Theorem provides a useful method to evaluate higher powers of binomial expressions ( e.g. a+b). It states that :

$$\displaystyle (a+b)^n = C(n, 0) a^n b^0 + C(n, 1) a^{n-1} b^1 + C(n, 2) a^{n-2} b^2 + \dots + C(n, n) a^{0} b^n \\$$

When any one of the two terms, say a = 1

$$\displaystyle (1+b)^n = C(n, 0) + C(n, 1) b + C(n, 2) b^2 + \dots + C(n, n)b^n \\$$

{eq}{/eq}

Given function :

$$\displaystyle f(x) = (1 + 2 x)^4 \\$$

Here, a = 1

b = 2x

n = 4

Therefore, we can expand f(x) as :

{eq}\displaystyle{f(x) = (1 + 2 x)^4 \\ \Rightarrow f(x) = C(4, 0) 1^n (2x)^0 + C(4, 1) 1^{n-1} (2x)^1 + C(4, 2) 1^{n-2} (2x)^2 + C(4, 3) 1^{n-3} (2x)^3 + C(4, 4) 1^{n-4} (2x)^4 \\ \Rightarrow f(x) = 1 \times 1 \times 1 + 4 \times 1 \times 2x + 6 \times 1 \times 4x^2 + 4 \times 1 \times 8x^3 + 1 \times 1 \times 16x^4 \\ \Rightarrow f(x) = 1 + 8x + 24x^2 + 32x^3 + 16x^4 \\ } {/eq}