Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (x-2 y)^{3} $$

Step-by-Step Solution

Answer

The polynomial is \(x^3 - 6x^2y + 12xy^2 - 8y^3\).

Step 1: Recognize the Binomial Expansion Formula

We can expand the cube of a binomial \((a-b)^3\) using the binomial theorem formula: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). In this problem, \(a = x\) and \(b = 2y\).

Step 2: Apply the Formula to x and 2y

Substitute \(a = x\) and \(b = 2y\) into the formula: \((x - 2y)^3 = x^3 - 3(x^2)(2y) + 3(x)(2y)^2 - (2y)^3\).

Step 3: Simplify Each Term

Calculate each term one by one:1. \(x^3\) remains as is. 2. \(-3(x^2)(2y) = -6x^2y\).3. \(3(x)(2y)^2 = 3x(4y^2) = 12xy^2\).4. \(-(2y)^3 = -8y^3\).

Step 4: Combine the Simplified Terms

Now, write out the expanded polynomial by combining all simplified terms. This gives us: \(x^3 - 6x^2y + 12xy^2 - 8y^3\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra that allows us to expand expressions of the form \((a+b)^n\), where \(n\) is a non-negative integer. In simple words, it helps to neatly write out the power of a sum. This theorem is particularly powerful because it not only tells us how to expand these expressions but also provides a coefficient for each term. For instance, in the expansion of \((a-b)^3\), we use the formula: - \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)To use the binomial theorem, follow these steps:- Identify the values of \(a\) and \(b\). For example, if you have \((x-2y)^3\), then \(a = x\) and \(b = 2y\).- Apply the pattern given by the theorem to expand the expression.Thus, understanding the Binomial Theorem is crucial for handling polynomials efficiently and effectively.

Cubic Polynomials

Cubic polynomials, as the name suggests, involve terms raised to the third power. They play an integral part in many mathematical applications. Generally, a cubic polynomial is written in the form: \[ ax^3 + bx^2 + cx + d \]where \(a\), \(b\), \(c\), and \(d\) are constants and \(a eq 0\). In our specific exercise, we expanded \((x-2y)^3\), resulting in a polynomial with each of its components having at least one factor raised to the third degree or a product that adds up to the cube. Here are characteristics to keep in mind:- The maximum degree of any term is 3.- They're often used to represent real-world scenarios like volume calculations due to their three-dimensional nature.When expanding, it's key to track coefficients and powers carefully, ensuring each term is simplified to reveal the polynomial's true structure.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that helps to present equations or formulas in the most efficient form. After expanding a binomial, as demonstrated in the step-by-step solution of \((x-2y)^3\), simplification means breaking down each component until no further simplification is possible:- Evaluate each term: Start by handling individual parts. For the expression \(-3(x^2)(2y)\), calculate the multiplication inside, resulting in \(-6x^2y\).- Combine like terms: Though the example had distinct terms, in broader contexts, combining terms with the same powers can further simplify the expression.- Correctly apply arithmetic operations in each term, ensuring accuracy in negative signs and coefficients.The ultimate goal is to make expressions neat and manageable, unlocking a clearer understanding of underlying algebraic concepts.