Problem 56 - College Algebra

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: Understand the Binomial Theorem

The Binomial Theorem tells us how to expand expressions of the form \( (a + b)^n \), where \( n \) is a natural number. For \( (x - 1)^3 \), this means expanding out \( (x - 1)(x - 1)(x - 1) \).

Step 2: Perform Multiplication

Start by multiplying the first two expressions, \( (x - 1)(x - 1) \), and then, multiply the result by the third expression. Therefore, done incrementally: First, \( (x - 1)(x - 1) = x^2 - 2x + 1 \), then we multiply this by \( (x - 1) \), which gives \( (x^2 - 2x + 1)(x - 1) \).

Step 3: Distribution of terms

Distribute terms by multiplying each term in the expression \( (x^2 -2x +1) \) by \( (x-1) \). This gives us \( x^3 - 2x^2 + x - x^2 + 2x - 1 = x^3 - 3x^2 + 3x - 1 \).