Problem 66 - Algebra for College Students

Question

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{x^{2}-(y-1)^{2}}{(y-1)^{2}-x^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-1\).

Step 1: Recognize the Expression Type

The given expression \(\frac{x^{2}-(y-1)^{2}}{(y-1)^{2}-x^{2}}\) is a difference of squares in both the numerator and the denominator. It can be rewritten using the formula for a difference of squares: \(a^{2} - b^{2} = (a + b)(a - b)\).

Step 2: Factorize the Numerator

Factorize the numerator \(x^{2}-(y-1)^{2}\) using the difference of squares: \((x + (y - 1))(x - (y - 1))\). This simplifies to \((x + y - 1)(x - y + 1)\).

Step 3: Factorize the Denominator

Factorize the denominator \((y-1)^{2} - x^{2}\) using the difference of squares: \((y-1 + x)(y-1 - x)\). This can be rewritten as \((x + y - 1)(x - y + 1)\) with a sign change.

Step 4: Simplify the Rational Expression

The expression now appears as \(\frac{(x + y - 1)(x - y + 1)}{-(x + y - 1)(x - y + 1)}\). The terms \((x + y - 1)\) and \((x - y + 1)\) cancel each other out, leaving \(-1\) as the simplified expression.

Key Concepts

Difference of Squares

When dealing with algebraic expressions, the **difference of squares** is a powerful tool for simplification. This special product occurs when you have two squared terms separated by a subtraction sign, like in our expression: \(x^2 - (y - 1)^2\). The general formula is \(a^2 - b^2 = (a + b)(a - b)\). Notice how it factors into two binomials. This structure is essential because it allows us to break down more complex algebraic expressions into simpler factors.
For example, using this on our original expression, we recognized both the numerator and the denominator as differences of squares. This recognition is crucial because it guides us on how to factor them.

• The numerator, \(x^2 - (y - 1)^2\), simplifies to \((x + (y - 1))(x - (y - 1))\).
• The denominator, \((y-1)^2 - x^2\), is simplified similarly, which shows the versatility and importance of this technique.

Understanding and applying the difference of squares can make solving equations and simplifying expressions much easier.

Algebraic Expressions

**Algebraic expressions** are combinations of numbers, variables, and mathematical operations. They form the building blocks of algebra and are used to represent real-world problems and more complex equations.
In our context, the rational expression \(\frac{x^2 - (y-1)^2}{(y-1)^2 - x^2}\) is made up of algebraic expressions in both the numerator and the denominator. Recognizing these components, such as squares or linear terms, helps in applying the right mathematical rules.

• Variables like \(x\) and \(y\) allow expressions to represent a range of numbers.
• Operations like subtraction can transform expressions significantly, introducing structures like differences of squares.
• Factoring techniques arise as solutions in manipulating these expressions to simplify or solve algebraic equations.

Having a strong understanding of algebraic expressions is key to figuring out how to simplify and manipulate these types of mathematical problems efficiently.

Factoring

**Factoring** is the process of rewriting an expression as a product of its factors. This technique is especially useful in simplifying complex expressions, solving equations, and finding zeros of functions.

• In our expression, \(x^2 - (y-1)^2\), factoring involves recognizing it as a difference of squares, allowing us to use the formula \((a+b)(a-b)\).
• For the denominator \((y-1)^2 - x^2\), a similar factoring technique—combined with attention to sign changes—simplifies it further.

Factoring transforms our expression from something cumbersome into simpler factors, making it easier to interpret and solve. The resulting simplification leads to expressions that are easier to cancel out or compute, such as the transformation from the complex fraction to the simplified -1. Understanding how factoring works and applying it to algebraic expressions helps ease the problem-solving journey and opens the door to solving more advanced mathematical challenges.