Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated operations and simplify. \(\frac{x^{2}-x-6}{x-3}\)

Step-by-Step Solution

Answer

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

Step 1: Factor the Numerator

The expression in the numerator is a quadratic \[ x^2 - x - 6. \] To factor this, look for two numbers that multiply to \(-6\) and add to \(-1\). Those numbers are \(-3\) and \(2\). Thus, the factored form of the numerator is \((x-3)(x+2)\).

Step 2: Simplify the Fraction

The original expression was \[ \frac{x^{2}-x-6}{x-3}. \] Substituting the factored numerator from Step 1, we have: \[ \frac{(x-3)(x+2)}{x-3}.\] Observe that \((x-3)\) is common in the numerator and the denominator. Thus, we can cancel \((x-3)\) from both the numerator and the denominator, simplifying the expression to \(x+2\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is an essential algebraic skill. It involves rewriting a quadratic expression as a product of its linear factors. The general form of a quadratic is \( ax^2 + bx + c \). To factor it, you need to find two numbers that multiply to \( ac \) and add up to \( b \).
For the expression \( x^2 - x - 6 \), you need numbers that multiply to \(-6\) (the product of \( 1 \cdot -6 \)) and add to \(-1\).

• In this case, the numbers are \(-3\) and \(2\).
• Therefore, \( x^2 - x - 6 \) factors into \((x-3)(x+2)\).

Factoring transforms the quadratic into a simpler form that reveals its roots. In this example, the roots are \( x = 3 \) and \( x = -2 \).
If a quadratic expression can be factored, it becomes significantly simpler to work with, especially in the context of rational expressions.

Cancelling Common Factors

Once you have factored the quadratic expression, it's time to simplify the fraction by cancelling common factors. A rational expression is essentially a fraction where the numerator and the denominator are polynomials.
If both the top and the bottom terms have common factors, they can be cancelled out to simplify the expression.
In our expression, \( \frac{(x-3)(x+2)}{x-3} \):

• The factor \((x-3)\) appears in both the numerator and the denominator.
• By cancelling the \( (x-3) \) term, you reduce the fraction to \( x + 2 \).

Remember, you can only cancel factors, not terms. Ensure that you simplify correctly to maintain the integrity of the expression. Simplification through cancellation makes computation easier and is a critical step in manipulating rational expressions.

Polynomial Division

Polynomial division isn't always straightforward like regular division. But when one polynomial is the factor of another, division is relatively simple. In dividing polynomials, the key is to look for a common polynomial factor in both the numerator and the denominator.
For the expression \( \frac{x^2-x-6}{x-3} \), once you factor the numerator, it can be seen as:

• \( \frac{(x-3)(x+2)}{x-3} \)

Here, \( (x-3) \) divides both itself in the numerator and the denominator directly. Since the numerator is a multiple of the denominator (after factoring), you can perform a simple division:

• Cancel \( (x-3) \) from both, leaving \( x+2 \).

This straightforward division only works if the term you are dividing by is a factor of the numerator as well. Polynomial division allows you to reach simpler results quickly, and it gives insight into the function's behavior, such as identifying any restrictions coming from cancelling terms.