Problem 70 - Algebra 2

Question

Expand each binomial. $$ (x-y)^{3} $$

Step-by-Step Solution

Answer

The expanded form of \((x-y)^3\) is \(x^3 - 3x^2y + 3xy^2 - y^3\).

Step 1: Binomial theorem application

Use the binomial theorem, which states that \((a-b)^n = \sum_{k=0}^n (-1)^k C(n, k) a^{n-k} b^{k}\), where \(C(n, k)\) is the combination of \(n\) and \(k\), to expand \((x-y)^3\).

Step 2: Coefficient calculation

Calculate the coefficients using combination formula. The combinations are as follows: \(C(3,0)=1, C(3,1)=3, C(3,2)=3, C(3,3)=1\).

Step 3: Expansion arrangement

Now substitute these combinations back into the equation along with the corresponding powers of \(x\) and \(y\), which should give the expanded binomial as \(x^3 - 3x^2y + 3xy^2 - y^3 \).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that provides a way to expand expressions raised to a power, particularly those of the form \((a - b)^n\). It tells us how to break down such an expression into a sum of terms involving coefficients, powers of the first term, and powers of the second term. The formula is: \[(a - b)^n = \sum_{k=0}^n (-1)^k C(n, k) a^{n-k} b^{k}\]Here:

• \(n\) is the non-negative integer exponent.
• \(C(n, k)\) represents the binomial coefficients, which indicate how many ways \(k\) elements can be chosen from \(n\) elements.
• The alternating \((-1)^k\) accounts for the subtraction within the binomial.

This theorem simplifies the expansion process by summarizing it into manageable steps, making it easier to handle larger exponents.

Coefficient Calculation

Calculating coefficients is a crucial part of the binomial expansion. Coefficients help determine the weight of each term in the expansion. The binomial coefficient, \(C(n, k)\), is calculated using the formula:\[C(n, k) = \frac{n!}{k!(n-k)!}\]For example, when expanding \((x-y)^3\), we need to calculate:

• \(C(3,0) = \frac{3!}{0!3!} = 1\)
• \(C(3,1) = \frac{3!}{1!2!} = 3\)
• \(C(3,2) = \frac{3!}{2!1!} = 3\)
• \(C(3,3) = \frac{3!}{3!0!} = 1\)

These calculated coefficients dictate how many times each term in the expansion will appear and define the structure of the expanded formula, ultimately leading to the expression \(x^3 - 3x^2y + 3xy^2 - y^3\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They form the building blocks for more complex mathematical ideas like equations and functions. When expanding binomials, as seen in the expression \(x^3 - 3x^2y + 3xy^2 - y^3\), we are essentially evaluating an algebraic expression in terms of another.
Rewriting complex expressions in expanded form allows math practitioners to:

• Clearly identify and differentiate terms and their coefficients.
• Better understand how variables interact when raised to powers.
• Simplify and solve equations more easily, due to the clearer picture of term distribution.

Whether simplifying, expanding, or factoring, grasping how algebraic expressions operate leads to proficiency in various areas of mathematics.