Problem 58 - Elementary and Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$ \frac{4(t-1)+4}{4 t+4} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{t}{t+1}\).

Step 1: Simplify the Numerator

First, simplify the numerator, which is given as \(4(t-1) + 4\). Distribute the \(4\) in the expression: \(4 \cdot t - 4 \cdot 1 + 4\). This simplifies to \(4t - 4 + 4\) which further simplifies to \(4t\).

Step 2: Simplify the Denominator

Next, simplify the denominator which is \(4t + 4\). This expression is already simplified.

Step 3: Cancel Common Factors

Now that the numerator is \(4t\) and the denominator is \(4t + 4\), we look for a common factor to cancel out. Factor \(4\) out of the denominator: \( 4(t + 1)\). Factor \(4\) from the numerator as well: \(4(t)\).

Step 4: Simplify the Fraction

Now write the fraction as \(\frac{4t}{4(t + 1)}\). You can see that \(4\) can be canceled out from the numerator and the denominator, since it is a common factor. The expression simplifies to \(\frac{t}{t + 1}\).

Key Concepts

Factoring Expressions

Factoring expressions is a key concept in algebra that involves breaking down an expression into simpler components, often to simplify or solve problems. In our exercise, the numerator and denominator are both expressions that can be factored. Factoring is the process of rewriting an expression as a product of its factors.

• In the numerator, \(4(t-1) + 4\), notice how we distribute and rearrange terms so we can factor out the common factor.
• After the distribution, we simplify to get \(4t\), which can be expressed as \(4 \cdot t\).

Factoring helps in managing complex expressions by breaking them down into their simplest components. This opening step is crucial as it sets up further simplification down the line. If you can manage to spot common factors quickly, you can make short work of simplifying seemingly complex algebraic fractions.

Simplifying Rational Expressions

Simplifying rational expressions is akin to simplifying fractions, but we work with expressions in the numerators and denominators instead of mere numbers. Our original expression \[ \frac{4(t-1)+4}{4t+4} \]needed simplification for easier handling.
Here’s how we simplified the expression:

• Simplify the numerator to eliminate complex components and then identify any common factors. In this case, simplifying \(4(t-1) + 4\) led us to \(4t\).
• Address the denominator in the same way, ensuring it remains in its simplest form so you can compare it easily with the simplified numerator.

By removing common factors of \(4\) from the fraction, it shrinks down to \(\frac{t}{t+1}\). Always remember to ensure that you do not divide terms by variables that could nullify the meaning of the expression by turning the denominator into zero.

Common Factors in Algebra

Common factors in algebra play a pivotal role in simplifying expressions. These factors are terms that appear in both the numerator and the denominator of a fraction, and when identified, can be cancelled to simplify the expression. In our problem, both the numerator and denominator have 4 as a common factor.

• Look closely at both the numerator \(4t\) and the denominator \(4t + 4\); both can be divided by 4.
• Extracting the 4 reveals a simpler relationship between the remaining terms.

When you successfully identify and cancel out these factors, it doesn't change the value of the expression but certainly makes it easier to work with. Common factors should be the first thing to look for when aiming for the simplification of any algebraic expression.