Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{x^{2}-x-6}{x-3} $$

Step-by-Step Solution

Answer

The simplified result is \(x + 2\).

Step 1: Factor the Numerator

The numerator of the fraction is a quadratic expression: \(x^2 - x - 6\). We can factor this into two binomials. Looking for two numbers that multiply to \(-6\) and add to \(-1\), we find \(-3\) and \(2\). So, the factoring is \((x - 3)(x + 2)\).

Step 2: Write the Fraction with Factored Numerator

Substitute the factored form of the numerator into the original fraction. We get: \(\frac{(x - 3)(x + 2)}{x - 3}\).

Step 3: Simplify the Fraction by Canceling Common Factors

Observe that \(x - 3\) is present in both the numerator and the denominator. Cancel \(x - 3\) from the numerator and the denominator, leaving us with \(x + 2\).

Step 4: State the Simplified Result

After canceling the common factor, the expression simplifies to \(x + 2\).

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill in algebra that helps to simplify expressions by breaking them down into simpler components. A quadratic expression is one that has a degree of two, such as \(x^2 - x - 6\). To factor a quadratic expression into two binomials, you need to find two numbers that *multiply* to give the last term (constant) and *add* to give you the middle term's coefficient. In our example:

• We need numbers that multiply to \(-6\).
• These numbers must also add up to \(-1\).
• After considering possibilities, we find \(-3\) and \(2\) work.

Thus, we can express the quadratic as two binomials: \((x - 3)(x + 2)\). These binomials are the factors of the quadratic equation.Understanding this concept is crucial because it lays the foundation for solving quadratic equations and simplifying algebraic expressions.

Simplifying Fractions

Simplifying fractions makes an expression easier to understand and solve. It involves reducing the fraction to its simplest form with the smallest possible numerator and denominator.The initial fraction from the exercise was \(\frac{(x - 3)(x + 2)}{x - 3}\). A fraction is in its simplest form when the numerator and denominator are coprime, meaning they do not share any common factors other than 1.In our case:

• The numerator is already factored: \((x - 3)(x + 2)\).
• The denominator is \(x - 3\).
• Both contain the term \(x - 3\).

By simplifying this fraction, you make it easier to work with, reducing the complexity of subsequent calculations. It's particularly useful when dealing with more complex algebraic expressions.

Canceling Common Factors

Canceling common factors is the process of removing identical terms from both the numerator and the denominator of a fraction. This step is crucial in simplifying algebraic fractions.In the example \(\frac{(x - 3)(x + 2)}{x - 3}\), both the numerator and the denominator contain the factor \(x - 3\). Because of this, you can cancel \(x - 3\) from the numerator and the denominator, simplifying the expression to \(x + 2\).Remember:

• You can only cancel factors, not individual terms in addition or subtraction. Both parts of the fraction must have exactly the same factor.
• Canceling simplifies the expression to its simplest form, making it easier to interpret.

This technique is fundamental in algebra and essential for working efficiently with equations and expressions, as it reduces complexity and aids in solving problems more easily.