Problem 51 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The product of \((x+1)^3\) is \(x^3+3x^2+3x+1\).

Step 1: Expand the Cubic Binomial

The given expression is a cubic binomial, meaning the entire expression \((x+1)\) is being cubed. Apply the binomial theorem which is \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^k\). The expression \((x+1)^3\) translates to \((x+1)(x+1)(x+1)\). When expanded this becomes \(x^3+3x^2+3x+1\). The coefficients 1, 3, 3, 1 corresponds to the fourth row of the Pascal's Triangle, which is a simple way to determine coefficients in binomial expansions.

Step 2: Final Simplification

Combine any like terms in the expression if applicable. However, in this case, \(x^3+3x^2+3x+1\) does not contain like terms, so the expression is already in its simplest form.