Problem 16 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$x^{2}-(y-5)^{2}$$

Step-by-Step Solution

Answer

Factor: \((x - y + 5)(x + y - 5)\).

Step 1: Identify the Difference of Squares Pattern

The difference-of-squares pattern is expressed as \(a^2 - b^2 = (a - b)(a + b)\). In this problem, compare \(x^2 - (y-5)^2\) with the pattern to identify \(a\) and \(b\). Here, \(a = x\) and \(b = y - 5\).

Step 2: Apply the Pattern

Substitute \(a = x\) and \(b = y - 5\) into the difference-of-squares formula. This gives \((x - (y - 5))(x + (y - 5))\).

Step 3: Simplify the Expression

Simplify the expression by removing the parentheses: \((x - (y - 5))\) becomes \((x - y + 5)\) and \((x + (y - 5))\) becomes \((x + y - 5)\). Thus, the factorized form is \((x - y + 5)(x + y - 5)\).

Key Concepts

Factoring Polynomials

When we talk about factoring polynomials, we refer to the process of breaking down a polynomial into simpler terms or factors. Factoring is much like taking a number apart into its prime factors. The main goal is to express a polynomial as a product of its factors, which are easier to handle and analyze.
Polynomials can be quadratic, cubic, or of higher degrees, and each follows different rules for factoring. One of the interesting and useful patterns in factoring is the difference-of-squares.

• The pattern of the difference of squares is expressed as \(a^2 - b^2 = (a - b)(a + b)\).
• This pattern emerges when you have two terms, both of which are perfect squares, separated by a subtraction symbol.

Applying this pattern simplifies many algebraic expressions and solves equations efficiently. In our exercise, recognizing \(x^2 - (y-5)^2\) as a difference of squares allows us to factor it into simpler binomials.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. These expressions represent mathematical relationships and are fundamental in algebra.
In our given problem, the expression \(x^2 - (y-5)^2\) consists of:

• The variable \(x\), squared represents the area of a square with side \(x\).
• The expression \((y-5)^2\) is another square, where the side is \(y-5\).

Recognizing the structure of algebraic expressions helps in identifying patterns such as the difference of squares. Understanding each component is key to simplifying, factoring, and solving equations. It forms the backbone for operations in algebra and leads to developing more complex mathematical theories and solutions.

Patterns in Mathematics

Patterns in mathematics are recognizable sequences or structures that help simplify complex problems. They play a crucial role in problem-solving and are especially important in algebra. Recognizing patterns allows mathematicians and students alike to apply standardized methods to seemingly complicated problems.
In this exercise, identifying the difference-of-squares pattern is instrumental.

• This pattern allows us to break down the expression \(x^2 - (y-5)^2\) effortlessly.
• It opens the path to factoring many similar algebraic expressions.

By learning and mastering patterns such as this, students can improve their mathematical fluency and problem-solving speed. Patterns help to uncover the hidden simplicity in many mathematical challenges and enhance the overall understanding of the subject.