Problem 64 - Elementary and Intermediate Algebra

Question

Factor. \(1-(x+y)^{3}\)

Step-by-Step Solution

Answer

The expression factors to \((1-(x+y))(1 + (x+y) + x^2 + 2xy + y^2)\).

Step 1: Identify the Formula to Use

The expression \( 1 - (x+y)^3 \) fits the pattern of a difference of cubes. The difference of cubes formula is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). Here, we identify \( a = 1 \) and \( b = (x+y) \).

Step 2: Apply the Difference of Cubes Formula

Substitute \( a = 1 \) and \( b = (x+y) \) into the difference of cubes formula: \[ 1^3 - (x+y)^3 = (1 - (x+y))(1^2 + 1(x+y) + (x+y)^2) \].

Step 3: Simplify the Expression

First, simplify the terms in the formula: \( 1^2 = 1 \), \( 1(x+y) = (x+y) \), and \((x+y)^2 = x^2 + 2xy + y^2 \).

Step 4: Write the Final Factored Form

Substituting the simplified terms into the difference of cubes expression gives: \[ (1-(x+y))(1 + (x+y) + x^2 + 2xy + y^2) \]. This is the fully factored form of the original expression.

Key Concepts

Understanding the Difference of Cubes

The difference of cubes is a type of polynomial expression that takes the form \( a^3 - b^3 \). This specific pattern can be factored using a special formula:

• \((a-b)(a^2 + ab + b^2)\)

This formula is useful because it provides a way to break down a seemingly complex cubic expression into a simpler, more manageable product of terms.
For the expression \(1 - (x+y)^3\), it fits this "difference of cubes" pattern by setting \(a = 1\) and \(b = (x+y)\). By substituting these into the formula, we turn \(1^3 - (x+y)^3\) into a product of polynomials that is easier to work with for further operations or simplifications.
Recognizing when an algebraic expression is a difference of cubes is key because it opens the door to simplify the expression and find its roots or solutions more easily.

Exploring Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are like a mathematical sentence that conveys an operation or series of operations. Each part of an expression is called a term, such as \((x+y)^3\) in the expression \(1 - (x+y)^3\).
An algebraic expression can be simple, like \(x + 2\), or complex, involving more operations and terms like the one in the exercise. Understanding how to manipulate and factor these expressions is crucial for solving equations in algebra.

• Expressions are constructed using addition, subtraction, multiplication, and division.
• They can include parentheses to show grouping.
• Variables within the expressions can represent unknown numbers.

Mastering how to handle different forms of algebraic expressions, such as identifying patterns like the difference of cubes, greatly aids in simplifying and solving equations.

Digging into Polynomials

Polynomials are algebraic expressions that consist of variables raised to varying powers and having coefficients. They can be classified by degree depending on the highest power of the variable present.
In the expression \(1 - (x+y)^3\), we are dealing with a cubic polynomial due to the term \((x+y)^3\). This is characterized by:

• A highest degree of 3.
• Potential variable terms like \(x^3, y^2\), and combinations thereof.
• Constant terms such as \(1\).

Factoring is a fundamental polynomial operation which breaks down the polynomial into simpler, more useful expressions.
Understanding the properties and behaviors of polynomials allows for more effective manipulation and simplification, which is invaluable when solving calculus problems, physics equations, and even real-world applications.