Problem 33 - Elementary and Intermediate Algebra

Question

Factor. \((x+y)^{2}-z^{2}\)

Step-by-Step Solution

Answer

\((x + y - z)(x + y + z)\)

Step 1: Recognize the Formula

The expression \((x+y)^2 - z^2\) fits the pattern of a difference of squares. The difference of squares formula states that \(a^2 - b^2 = (a-b)(a+b)\). In this problem, \(a\) corresponds to \((x+y)\) and \(b\) corresponds to \(z\).

Step 2: Apply the Difference of Squares Formula

Substitute \(a = (x+y)\) and \(b = z\) into the difference of squares formula. This yields: \[(x+y)^2 - z^2 = ((x+y) - z)((x+y) + z)\].

Step 3: Simplify the Expression

Simplify the factored expression: \[((x+y) - z)((x+y) + z) = (x + y - z)(x + y + z)\]. This is the final, factored form of the expression.

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic pattern and it is quite useful when factoring expressions. This pattern occurs when you have two squared terms that are subtracted from each other.
This can be recognized as:

• One term is a perfect square, such as \(x^2\) or \(y^2\).
• There is a subtraction sign between these perfect squares to indicate the 'difference'.

The general formula for the difference of squares is given by:\[ a^2 - b^2 = (a-b)(a+b) \]This formula allows us to factor the expression into two binomials. It is very useful in algebra to simplify and solve equations. In the given problem, \( (x+y)^2 - z^2 \) follows this exact pattern. Recognizing this pattern is the key first step in simplifying such expressions.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers linked by mathematical operations such as addition, subtraction, multiplication, and division. In these expressions, variables are used to represent unknown or changeable values. In the provided problem, the expression \( (x+y)^2 - z^2 \) is an example of an algebraic expression. Here, we identify that variables \(x\), \(y\), and \(z\) can represent any number.Key components of algebraic expressions include:

• Terms: Parts of an expression separated by plus or minus signs.
• Constants: Numbers that do not change.
• Coefficients: Numbers multiplying the variables.

Understanding how to manipulate algebraic expressions is a cornerstone of algebra and helps in solving problems, finding unknowns, and simplifying expressions.

Factoring Techniques

Factoring is the process of breaking down an expression into a product of simpler expressions or factors. This technique is essential in algebra because it simplifies expressions. It makes solving equations and understanding relationships between variables much easier. In algebra, there are several factoring techniques. Some common methods include:

• Finding a common factor: Look for numbers or variables that are common across all terms.
• Grouping: Used when an expression has four or more terms.
• Difference of squares: As discussed, used when you encounter two perfect squares separated by a subtraction sign.

The given expression \( (x+y)^2 - z^2 \) demonstrates how the difference of squares technique is employed to factor into \[ (x + y - z)(x + y + z) \]. Being familiar with these methods is crucial for students to tackle various algebraic expressions effectively.