Problem 9 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x+2)^{3} $$

Step-by-Step Solution

Answer

\((x+2)^{3}= x^{3} + 6x^{2} + 12x + 8\)

Step 1: Understanding the Binomial Theorem

The Binomial Theorem states that \((a+b)^{n}= \sum _{k=0}^{n}{n \choose k}a^{n-k}b^{k}\). So, to expand \((x+2)^{3}\), we need to use \(a=x\), \(b=2\), and \(n=3\) in this formula.

Step 2: Applying the Binomial Theorem

Applying the theorem, we get: \((x+2)^{3} = {3 \choose 0}x^{3}2^{0} + {3 \choose 1}x^{2}2^{1}+ {3 \choose 2}x^{1}2^{2}+ {3 \choose 3}x^{0}2^{3}\).

Step 3: Simplifying the expression

We can simplify the expression by calculating the binomial coefficients \({3 \choose k}\) and powers of x and 2. This yields: \(x^{3} + 6x^{2} + 12x + 8\).