Problem 20 - Contemporary Precalculus

Question

Expand and (where possible) simplify the expression. $$(x-1)^{5}$$

Step-by-Step Solution

Answer

Question: Expand and simplify the expression \((x-1)^5\). Answer: \((x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1\)

Step 1: Identify a, b, and n

In our expression, we have: - a = x - b = -1 - n = 5

Step 2: Apply the binomial theorem

Using the binomial theorem formula, we can write the expression as a sum of terms: $$(x-1)^5 = \sum_{k=0}^5 \binom{5}{k} x^{5-k}(-1)^k$$

Step 3: Calculate each term

Calculate each term in the sum by replacing k with 0, 1, 2, 3, 4, and 5: - k=0: \(\binom{5}{0} x^{5}(-1)^{0} = 1 \cdot x^{5} \cdot 1 = x^{5}\) - k=1: \(\binom{5}{1} x^{4}(-1)^{1} = 5 \cdot x^{4} \cdot (-1) = -5x^{4}\) - k=2: \(\binom{5}{2} x^{3}(-1)^{2} = 10 \cdot x^{3} \cdot 1 = 10x^{3}\) - k=3: \(\binom{5}{3} x^{2}(-1)^{3} = 10 \cdot x^{2} \cdot (-1) = -10x^{2}\) - k=4: \(\binom{5}{4} x^{1}(-1)^{4} = 5 \cdot x^{1} \cdot 1 = 5x^{1}\) - k=5: \(\binom{5}{5} x^{0}(-1)^{5} = 1 \cdot x^{0} \cdot (-1) = -1\)

Step 4: Combine the terms

Combine all the terms calculated in step 3: $$(x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1$$ This is the expanded and simplified form of \((x-1)^5\).

Key Concepts

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial as a sum of distinct terms. Each term represents a different combination of the binomial components, raised to varying powers. In the expression \((x-1)^{5}\), we used the binomial theorem to expand it as a polynomial.

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form \((a+b)^{n}\). This expansion results in a polynomial, consisting of several terms where each term follows a specific pattern. Each term involves the original binomial components \(a\) and \(b\), raised to decreasing and increasing powers, respectively.

• In our example, \(a=x\), \(b=-1\), and \(n=5\).
• Each expanded term is also accompanied by a coefficient determined by combinatorial factors.

This systematic approach breaks down the expression into manageable parts, making it easier to simplify and understand. The result is a polynomial that expresses the same mathematical relationships as the original binomial raised to a power.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of elements. When we expand a binomial using the binomial theorem, combinatorial coefficients play a key role.

These coefficients, known as binomial coefficients, are calculated using combinations, denoted by \(\binom{n}{k}\). This notation represents the number of ways to choose \(k\) items from \(n\) items without considering the order. In the context of binomial expansion:

• They determine the multiplicative factor for each term.
• They can be found in Pascal's Triangle, where each number is the sum of the two numbers directly above it.

By applying binomial coefficients, each term in our expansion \((x-1)^{5}\) becomes a straightforward calculation. For instance, when \(k=2\), we calculate \(\binom{5}{2}\) to determine the third term’s coefficient. Understanding these coefficients is essential for efficient polynomial expansion and simplification.

Algebra

Algebra serves as the foundational framework for manipulating and solving expressions like polynomials. The step-by-step expansion of \((x-1)^5\) shows how algebraic principles allow us to systematically handle complex expressions.

Key Algebraic Activities:

• Substitution: Replacing \(k\) with specific values in the binomial expression to determine each term.
• Combining Like Terms: After expansion, terms with the same variable powers are summed to achieve a simplified form.
• Use of Negative Signs: Careful attention to the sign changes due to \((-1)^k\), affecting the terms positively or negatively.

The algebraic expansion of \((x-1)^5\) exemplifies the systematic way in which algebra untangles expressions, moving from a compact binomial to a detailed series of terms. This process not only simplifies but also reveals the underlying relationships within the original expression.