Problem 25 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-1)^{3}$$

Step-by-Step Solution

Answer

The expanded form of the binomial \((x-1)^{3}\) is \(x^{3} - 3x^{2} + 3x - 1\).

Step 1: Identify the Binomial and Power

The binomial to be expanded is \((x-1)\) and the power is \(3\). Write the binomial as \(a - b\), where \(a = x\) and \(b = 1\).

Step 2: Apply the Binomial Theorem

The Binomial Theorem is given by: \((a - b)^{n} = a^{n} - na^{n-1}b + \binom{n}{2}a^{n-2}b^{2} - \binom{n}{3}a^{n-3}b^{3} + \ldots\). Apply this formula to the problem, using \(x\) for \(a\), \(1\) for \(b\), and \(3\) for \(n\). This yields \((x - 1)^{3} = x^{3} - 3x^{2} + 3x - 1\).

Step 3: Simplify the Result

There are no like terms to combine, so the simplified form of the binomial expansion is \(x^{3} - 3x^{2} + 3x - 1\).

Key Concepts

Binomial Expansion

Understanding binomial expansion is critical for algebra and calculus students alike. It involves expressing a binomial power, such as \( (x - 1)^3 \), as a sum of terms. The Binomial Theorem provides a systematic method for expanding such expressions.

Consider a binomial \( (a+b)^n \). The theorem states that this expression can be expanded into a sum where each term has a specific form: coefficients from Pascal's Triangle, powers of \( a \), and powers of \( b \). The expanded form of \( (x-1)^3 \) is \( x^3 - 3x^2 + 3x - 1 \), derived from applying the theorem for \( n=3 \). Each term \( a^n, -na^{n-1}b, \ldots \) corresponds to the coefficients \( 1, 3, \ldots \) found in the 4th row of Pascal's Triangle (remember, counting starts at row 0).

A common exercise improvement advice is to reinforce the concept by solving additional problems, demonstrating alternative forms and connecting the theorem to other areas of mathematics such as combinatorics.

Algebraic Expressions

Algebraic expressions are the backbone of algebra and facilitate the representation of mathematical ideas. An expression like \( (x - 1)^3 \) is a concise way to describe the relationship between numbers and variables. When expanding algebraic expressions using the Binomial Theorem, it's crucial to identify the terms and their respective powers.

In the example \( (x - 1)^3 \), \( x \) takes the role of \( a \) and \( -1 \) takes the role of \( b \) in our theorem. Through expansion, we translate a compact binomial expression into a fully detailed sum that reveals all the individual terms. This can aid in further calculations, such as simplification or substitution. The process of translating these compact expressions into more explicit forms is an essential skill for solving more complex algebraic problems.

Combinatorics

Link to Binomial Theorem

Combinatorics, the branch of mathematics dealing with combinations and permutations, plays a significant role in understanding the coefficients of a binomial expansion. These coefficients are not random; they are binomial coefficients that count the number of ways to choose a certain number of elements from a larger set.

The coefficient \( \binom{n}{k} \) in the expansion signifies the number of ways to select \( k \) elements out of \( n \) options. For example, in our expansion of \( (x - 1)^3 \), the coefficients \( 1, 3, 3, 1 \) correlate to \( \binom{3}{0}, \binom{3}{1}, \binom{3}{2}, \binom{3}{3} \) respectively. These coefficients are key in solving problems involving probability, discrete structures, and other mathematical inquiries where the count of combinations is required.

By exploring combinatorial interpretation, students can visualize binomial expansion beyond mere algebraic manipulation, seeing it as a method to describe various combinations.