Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{t^{2}-4 t-21}{t+3} $$

Step-by-Step Solution

Answer

The simplified form is \( t - 7 \).

Step 1: Identify the Expression

The expression provided is \( \frac{t^2 - 4t - 21}{t + 3} \). Our goal is to simplify it by performing polynomial long division or factoring where possible.

Step 2: Factor the Numerator

The numerator is \( t^2 - 4t - 21 \). We need to factor it. We look for two numbers that multiply to -21 and add to -4. These numbers are -7 and 3, so we can factor the numerator as \((t - 7)(t + 3)\).

Step 3: Simplify by Cancelling Common Factors

The expression becomes \( \frac{(t - 7)(t + 3)}{t + 3} \). Since \(t + 3\) is a common factor in both the numerator and the denominator, we can cancel it out.

Step 4: Write the Simplified Expression

After cancelling the \( t+3 \) terms, the simplified expression is \( t - 7 \).

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding pieces of a puzzle that fit together perfectly. In polynomial division, particularly with rational expressions, the first step is to break the polynomial into simpler, more manageable parts. Here, we start with the quadratic expression in the numerator, \( t^2 - 4t - 21 \).

Our aim is to express it as a product of two binomials, just like turning a big problem into smaller, easier-to-solve problems. When factoring quadratics, we look for two numbers that multiply to the constant term (-21) and simultaneously sum up to the linear coefficient (-4).

These numbers are -7 and 3 because:

• -7 times 3 equals -21
• -7 plus 3 equals -4

Thus, we can rewrite our quadratic as \((t - 7)(t + 3)\). Breaking down quadratics like this simplifies the expression and prepares it for further cancellation.

Simplifying Expressions

Simplifying is all about reducing problems to their simplest form, like finding the most concise way to express an idea. Once our polynomial is factored, we can simplify the expression by canceling out common terms found in both numerator and denominator. Think of it as trimming off the superfluous parts.

In our example, we spot \((t + 3)\) in both the numerator and denominator. Since anything divided by itself equals one (except zero), we can cancel these terms out completely. This leaves us with only \( t - 7 \) in the numerator. Simplification reduces the expression to its essence, making it easier to interpret and use.

Algebraic Fractions

Algebraic fractions might seem intimidating at first, but they follow rules quite similar to numeric fractions. They involve the division of one polynomial by another. To handle them effectively, a clear understanding of both factoring and simplifying is crucial.

When you have a polynomial like \( \frac{t^2 - 4t - 21}{t + 3} \), it's essentially a fraction where the operations need to maintain balance. By factoring the numerator and simplifying, you ensure that the expression remains valid and consistent.

This process results in a cleaner, simpler form. With \( \frac{(t - 7)(t + 3)}{t + 3} \) simplified to \( t - 7 \), you've successfully simplified an algebraic fraction. Mastery of these operations allows for tackling more complex algebraic structures with confidence.