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Chapter 7: Problem 28

Simplify by removing a factor equal to 1. $$ \frac{3 a+12}{3} $$

Short Answer

Expert verified
The simplified form is \(a + 4\).

Step by step solution

01

Factor the Numerator

Identify common factors in the numerator. In the expression \(\frac{3a + 12}{3}\), both terms in the numerator have a common factor of 3. Factor out the 3: \(\frac{3(a + 4)}{3}\).
02

Cancel the Common Factors

Cancel the common factor in the numerator and the denominator. Both the numerator and the denominator have a factor of 3, so they cancel each other out, leaving: \(a + 4\).
03

Write the Simplified Form

After removing the common factor, the simplified expression is: \(a + 4\). Hence, the simplified form of the original expression \(\frac{3a + 12}{3}\) is \(a + 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is a key skill in algebra that helps in simplifying expressions. It involves breaking down a composite number into its prime factors or expressing a polynomial as a product of its factors. In our exercise, we start by factoring the numerator of the fraction \(\frac{3a + 12}{3}\).
Identifying common factors is the first step. Both 3a and 12 have 3 as a common factor.
This means we can factor out 3 from the numerator, making it \(\frac{3(a + 4)}{3}\).
It's like finding what's common in both terms and separating it out.
Getting comfortable with factoring in different scenarios (numbers, polynomials) makes handling complex algebraic expressions much easier.
Canceling Common Factors
Canceling common factors is a technique used to simplify fractions. After factoring, we look for common elements in both the numerator and the denominator.
In our example, the fraction became \(\frac{3(a + 4)}{3}\) after factoring.
Here, the numerator and the denominator both have a factor of 3.
To simplify this expression, we can cancel out the common factor, 3, from both the top and bottom.
This leaves us with the simplified form: \(a + 4\).
This technique is powerful because it reduces expressions to their simplest form, making them easier to handle and understand.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form. This often requires factoring and canceling common factors.
Let's review our original exercise to see this in action. We started with \(\frac{3a + 12}{3}\).
First, we factored the numerator to get \(\frac{3(a + 4)}{3}\).
Then, we canceled the common factor of 3 in both the numerator and the denominator.
The result was the simplified fraction, \(a + 4\).
Simplifying fractions makes complex expressions more manageable and easier to understand.
Always look for common factors and apply the process: Factor, cancel, and simplify. This consistent approach will make algebraic expressions much less intimidating.

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