Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated operations and simplify. \(\frac{t^{2}-4 t-21}{t+3}\)

Step-by-Step Solution

Answer

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

Step 1: Factor the Numerator

The first step is to factor the quadratic expression in the numerator \(t^2 - 4t - 21\). To do this, we look for two numbers that multiply to \(-21\) (the constant term) and add to \(-4\) (the coefficient of \(t\)). The numbers \(3\) and \(-7\) work because \(3 \times (-7) = -21\) and \(3 + (-7) = -4\). So, we can factor the numerator as \((t + 3)(t - 7)\).

Step 2: Simplify the Expression

After factoring, the expression becomes \(\frac{(t + 3)(t - 7)}{t + 3}\). Notice that there is a common factor of \(t + 3\) in both the numerator and the denominator. We can cancel this common factor.

Step 3: Write the Simplified Expression

After canceling the common factors, we are left with \(t - 7\). This is the simplified form of the original expression.

Key Concepts

Factoring Quadratics

Factoring quadratics is essential when simplifying algebraic expressions like fractions. Quadratic expressions often appear in the form of \( ax^2 + bx + c \). Our goal is to break them down, or "factor them," into simpler binomial expressions. This process makes it easier to identify and eliminate common factors with the denominator.

For quadratic expressions, the process involves:

• First, identifying two numbers that multiply to the constant term \( c \) and simultaneously add up to the coefficient \( b \).
• The factors of the example quadratic \( t^2 - 4t - 21 \) are 3 and -7. They multiply to -21 and add to -4.
• The factored form becomes \((t + 3)(t - 7)\).

Factoring is a powerful tool, revealing simplified relations hidden within complex algebraic structures. Understanding it improves problem-solving skills and can make algebra feel more accessible.

Simplifying Expressions

To simplify expressions, the key is to identify and eliminate common factors. Reducing the expression to its simplest form can make it easier to work with in future algebraic operations.

The process involves:

• First, after factoring, observe the expression: \( \frac{(t + 3)(t - 7)}{t + 3} \).
• Notice that \( t + 3 \) appears both in the numerator and the denominator.
• By canceling the common term \( t + 3 \), you are left with \( t - 7 \).

Canceling common factors is like simplifying fractions in arithmetic and results in a more concise expression. It's crucial, though, to ensure that cancellation is valid, meaning the factor isn't zero, as division by zero is undefined. This attention to detail leads to accurately simplified expressions.

Polynomial Division

Polynomial division is an operation that divides one polynomial by another, simplifying the expression just like long division in arithmetic.

In the case of algebraic fractions, polynomial division surfaces naturally when common factors are canceled out. Here's how:

• Consider \( \frac{(t + 3)(t - 7)}{t + 3} \): It's dividing a product by one of its factors.
• The division \( (t + 3) \) in the numerator by \( t + 3 \) in the denominator leaves us with the remaining factor \( t - 7 \).
• This is much like regular division; eliminating what's common leaves the rest of the quotient.

Polynomial division streamlines expressions and focuses on their essential parts. Understanding this can aid in tackling more challenging problems, where the simplification of fractions is just one necessary step in a wider solution.