Problem 12 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{x^{2}-3 x-10}{x^{2}-6 x+5} $$

Step-by-Step Solution

Answer

\(\frac{x + 2}{x - 1}\)

Step 1: Factor the Numerator

Start by factoring the quadratic expression in the numerator, which is \(x^2 - 3x - 10\). Look for two numbers that multiply to \(-10\) and add to \(-3\). The numbers \(-5\) and \(2\) work, so factor the numerator as: \((x - 5)(x + 2)\).

Step 2: Factor the Denominator

Factor the quadratic expression in the denominator, which is \(x^2 - 6x + 5\). Look for two numbers that multiply to \(5\) and add to \(-6\). The numbers \(-5\) and \(-1\) work, so factor the denominator as: \((x - 5)(x - 1)\).

Step 3: Simplify the Fraction

After factoring the numerator and denominator, the expression becomes: \(\frac{(x - 5)(x + 2)}{(x - 5)(x - 1)}\). Notice that \((x - 5)\) is a common factor in both the numerator and denominator, so it can be canceled out, leaving: \(\frac{x + 2}{x - 1}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to rewrite quadratic expressions as a product of two binomials. This is particularly useful when simplifying algebraic fractions, as it allows us to identify common factors in the numerator and denominator. To factor a quadratic expression like \(x^2 - 3x - 10\), you need to:

• Identify two numbers that multiply to the product of the constant term and the leading coefficient, here it is \(-10\).
• The same two numbers should add up to the linear coefficient \(-3\).

In this exercise, the numbers \(-5\) and \(2\) satisfy these conditions because \(-5 \times 2 = -10\) and \(-5 + 2 = -3\). Once you've found these numbers, you can rewrite the quadratic as \((x - 5)(x + 2)\). Remember, mastering this technique helps unravel more complex algebraic expressions.

Simplifying Expressions

Simplifying expressions is a key step in algebra to make complex problems more manageable. The goal is to reduce an expression to its simplest form. In this context, it involves canceling out common factors. Before simplifying, ensure that the expression is fully factored. From our example:When we factor \(x^2 - 3x - 10\) as \((x - 5)(x + 2)\) and \(x^2 - 6x + 5\) as \((x - 5)(x - 1)\), we identify \((x - 5)\) as a common factor in both the numerator and the denominator.

• Cancel \( (x - 5)\) from both the numerator and the denominator.
• The simplified expression becomes \( \frac{x + 2}{x - 1} \).

This doesn't just tidy up the expression; it also highlights the essential parts you need to focus on. Ensuring expressions are simplified is crucial for accurate and efficient problem-solving.

Rational Expressions

Rational expressions are fractions made up of polynomials, where the numerator and denominator are polynomial expressions. Simplifying rational expressions involves the same steps used for simplifying numerical fractions: factoring and canceling common terms.When dealing with rational expressions, always aim to:

• Factor both the numerator and the denominator completely.
• Identify and cancel out any common factors.

In our example, the rational expression \( \frac{x^2 - 3x - 10}{x^2 - 6x + 5} \) is simplified to \( \frac{x + 2}{x - 1} \) after canceling the common factor \((x - 5)\). Understanding these basic operations allows you to tackle more complicated algebraic fractions with confidence, ensuring that you properly manipulate and solve equations when necessary.