Problem 15 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (2 x+1)^{4} $$

Step-by-Step Solution

Answer

The binomial \((2x+1)^{4}\) simplified as per the Binomial Theorem is \(16x^4+32x^3+24x^2+8x+1\).

Step 1: Define Binomial Theorem

The Binomial Theorem states that \( (a + b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \). Here, \( \binom{n}{k} \) corresponds to the binomial coefficient which is equal to \( \frac{n!}{k! (n-k)!} \), where ! denotes the factorial operator.

Step 2: Apply Binomial Theorem

In the given equation \((2x+1)^{4}\), 'a' corresponds to 2x, 'b' corresponds to 1 and 'n' corresponds to 4. The binomial can be expanded as follows using the Binomial Theorem: \( (2x+1)^{4} = \sum_{k=0}^{4} \binom{4}{k} (2x)^{4-k} 1^{k} \)

Step 3: Simplify

On simplification, the equation turns into: \(16x^4+32x^3+24x^2+8x+1\)