Problem 25 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x-1)^{5} $$

Step-by-Step Solution

Answer

The expansion of \((x -1)^5\) using the Binomial Theorem is \(x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1\).

Step 1: Understand the Binomial Theorem Formula

The Binomial Theorem is \((a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} \cdot b^{k}\). This formula tells us that we can expand (a + b) raised to a power n using the binomial coefficients from Pascal's triangle, together with powers of a and b.

Step 2: Adjust the Formula to our Problem

In our case: \(a=x\), \(b=-1\), and \(n=5\), applying them to the binomial theorem, we get \((x - 1)^5 = \sum_{k=0}^{5} {5 \choose k} x^{5-k} \cdot (-1)^{k}\).

Step 3: Expansion and Simplification

Expand the expression given as follows: \((x - 1)^5 = {5 \choose 0} x^{5} \cdot (-1)^{0} + {5 \choose 1} x^{4} \cdot (-1)^{1} + {5 \choose 2} x^{3} \cdot (-1)^{2} + {5 \choose 3} x^{2} \cdot (-1)^{3} + {5 \choose 4} x^{1} \cdot (-1)^{4} + {5 \choose 5} x^{0} \cdot (-1)^{5} \), which simplifies to \(x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 \).