Problem 18 - Algebra and Trigonometry

Question

Expand each expression using the Binomial Theorem. $$ (x-1)^{5} $$

Step-by-Step Solution

Answer

The expanded form is \[ x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 \]

Step 1: Understand the Binomial Theorem

The Binomial Theorem states: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. In this exercise, \(a = x\), \(b = -1\) and \(n = 5\).

Step 2: Recognize the binomial coefficients

Use the binomial coefficients \( \binom{n}{k} \). For \( \binom{5}{k} \), where \(k\) goes from 0 to 5: \[ \binom{5}{0}, \binom{5}{1}, \binom{5}{2}, \binom{5}{3}, \binom{5}{4}, \binom{5}{5} \]. These coefficients are 1, 5, 10, 10, 5, and 1 respectively.

Step 3: Apply the binomial coefficients in the theorem

Substitute \(a = x\) and \(b = -1\) into the binomial expansion: \[ (x-1)^5 = \binom{5}{0} x^{5} (-1)^{0} + \binom{5}{1} x^{4} (-1)^{1} + \binom{5}{2} x^{3} (-1)^{2} + \binom{5}{3} x^{2} (-1)^{3} + \binom{5}{4} x^{1} (-1)^{4} + \binom{5}{5} x^{0} (-1)^{5} \]

Step 4: Compute each term

Calculate each term: \[ \binom{5}{0} x^5 (-1)^0 = 1 \cdot x^5 \cdot 1 = x^5 \]\[ \binom{5}{1} x^4 (-1)^1 = 5 \cdot x^4 \cdot (-1) = -5x^4 \]\[ \binom{5}{2} x^3 (-1)^2 = 10 \cdot x^3 \cdot 1 = 10x^3 \]\[ \binom{5}{3} x^2 (-1)^3 = 10 \cdot x^2 \cdot (-1) = -10x^2 \]\[ \binom{5}{4} x^1 (-1)^4 = 5 \cdot x \cdot 1 = 5x \]\[ \binom{5}{5} x^0 (-1)^5 = 1 \cdot 1 \cdot (-1) = -1 \]

Step 5: Write the final expanded form

Combine all the terms: \[ x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 \]

Key Concepts

binomial expansion

Binomial expansion is a method used to expand expressions that are raised to a power. It's based on the Binomial Theorem which states that any binomial expression \(a + b\) raised to a power \(n\) can be expanded as: \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \( \binom{n}{k} \) are the binomial coefficients. This formula helps in breaking down complex polynomial expressions into simpler, manageable terms.
For example, the expression \( (x - 1)^5 \) can be expanded using the binomial theorem. In this expression, \( a = x \), \( b = -1 \), and \( n = 5 \). Following the formula, each term involves calculating a binomial coefficient and then multiplying it by the appropriate powers of \( x \) and \(-1\).
This method is especially useful in algebra to simplify expressions and solve equations involving higher powers of binomials.

binomial coefficients

where \( n! \) (n factorial) is the product of all positive integers up to \( n \). The binomial coefficients for \( (x - 1)^5 \) when fully expanded range from \( \binom{5}{0} \) to \ \binom{5}{5} \. These coefficients are 1, 5, 10, 10, 5, and 1 respectively.
Calculating each coefficient helps us understand the weight each term will have in the final polynomial expansion. Each binomial coefficient is crucial in determining the magnitude of each resulting term in the expansion.
Combining binomial coefficients with powers of the individual terms allows us to construct the expanded polynomial, stepping from numerical coefficients to powers of given variables.

algebraic expressions

Algebraic expressions are mathematical phrases involving variables, constants, and operations like addition, subtraction, multiplication, and division. In the realm of binomials, an example expression is \( (x - 1) \). When an algebraic expression like this is raised to a power, the binomial theorem helps in expanding it.
For instance, expanding \( (x - 1)^5 \) is a way to express it as a sum of simpler terms. Each term in an expanded form like \( x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 \) is an algebraic expression itself. These expansions allow for deeper insights into the behavior of polynomials and facilitate operations like differentiation, integration, or even polynomial long division.
The ability to manipulate and expand algebraic expressions lays foundational skills for problem-solving in algebra and higher mathematics.