Problem 51 - College Algebra

Question

Find each product. $$(x+1)^{3}$$

Step-by-Step Solution

Answer

The cubic of the expression \( (x+1) \) is \( x^{3} + 3x^{2} + 3x + 1 \).

Step 1: Write Binomial Expression

Write down the binomial expression whose cube needs to be found. This is the expression: \((x+1)^{3}\).

Step 2: Apply Binomial Theorem

Apply the binomial theorem, \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \), where \( n = 3 \), \( a = x \), \( b = 1 \), and \( \binom{n}{k} \) is the binomial coefficient: \((x+1)^{3} = \binom{3}{0}x^{3}1^{0} + \binom{3}{1}x^{2}1^{1} + \binom{3}{2}x^{1}1^{2} + \binom{3}{3}x^{0}1^{3}\) .

Step 3: Calculate Binomial Coefficients

For binomial coefficient, calculate as: \(\binom{n}{m} = \frac{n!}{m!(n-m)!}\), so, \(\binom{3}{0} = 1, \binom{3}{1} = 3, \binom{3}{2} = 3, \binom{3}{3} = 1 \). Replace these values.

Step 4: Simplify the Expression

On substituting the calculated coefficients and simplifying, we obtain: \(x^{3} + 3x^{2} + 3x + 1\).