Problem 32 - Calculus: One and Several Variables

Question

Expand \(g(x)\) as indicated. \(g(x)=(x-1)^{n} \quad\) in powers of \(x\).

Step-by-Step Solution

Answer

To expand the given function \(g(x) = (x-1)^n\) in powers of x using the binomial theorem, we have: \[g(x) = \sum_{k=0}^n {n \choose k} (-1)^k x^{n-k}\] This expression represents \(g(x)\) expanded in powers of \(x\).

Step 1: Apply the binomial theorem

We are given \(g(x) = (x-1)^n\). Let \(a = x\) and \(b = -1\). We can now apply the binomial theorem to expand this expression: \[g(x) = \sum_{k=0}^n {n \choose k} x^{n-k}(-1)^k\]

Step 2: Simplify the expression

Now we need to simplify this expression by incorporating the \((-1)^k\) term into the expansion, and then write \(g(x)\) in powers of \(x\). We have: \[g(x) = \sum_{k=0}^n {n \choose k} (-1)^k x^{n-k}\] This expression represents the function \(g(x)\) expanded in powers of \(x\).

Key Concepts

Expansion of expressions

Expansion of expressions refers to rewriting algebraic expressions in an extended form. Using the Binomial Theorem, we expand expressions like \[(x-1)^n\]into a sum of terms with different powers of \(x\). The formula usually involves combinations and powers. Understanding the basics is crucial before diving into more complex expression expansions.

• Identify components: Find base terms like \(a = x\) and \(b = -1\).
• Apply Binomial Theorem: Use it to form a series expansion.
• Repetitive process: Break down terms step-by-step.

By systematically applying this method, you can manage expressions of varying complexity and simplify solving problems in algebra or calculus.

Power functions

Power functions involve expressions with variables raised to a power, such as \(x^n\). When dealing with expansions, understanding the distribution of powers in each term is essential. Each term in an expanded expression has a power of \(x\) determined by the Binomial Theorem. This power decreases successively from the initial exponent \(n\) in the formula \[ {x}^{n-k} \]. Knowing how these power functions behave is important, as they give shape to the polynomial resulting from an expansion. Understanding power functions helps predict the expansion's behavior:- **Exponent Relation**: Each term's power relates to its position in the expansion.- **Decreasing Order**: Powers of \(x\) lessen as you move through the series.Recognizing how to distribute and simplify these powers helps unravel complex polynomials during expansion.

Combinatorial coefficients

Combinatorial coefficients, often expressed as \({n \choose k}\), also known as "n choose k," are a crucial part of expressions involving binomial expansions. They determine the coefficients that appear with each term in the expansion.These coefficients are derived from combinations, which tell us how many ways we can select \(k\) elements from \(n\) total elements, following the formula:\[ {n \choose k} = \frac{n!}{k!(n-k)!} \]The combinatorial coefficients assign weight to each term, indicating its relative contribution to the polynomial. Here’s how they work in practice:

• **Number of Terms**: Given \(n\), the expansion has \(n+1\) terms.
• **Coefficient Calculation**: Apply the formula to compute each coefficient for terms \(k=0\) to \(n\).
• **Symmetry Property**: Coefficients are symmetric, meaning \({n \choose k} = {n \choose n-k}\).

By mastering these coefficients, you unlock the power to delve into and manipulate binomial expansions with ease.