Problem 35 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{3 x+6}{3 x}$$

Step-by-Step Solution

Answer

\(\frac{x+2}{x}\)

Step 1 - Identify the expression

The given rational expression is \(\frac{3x+6}{3x}\).

Step 2 - Factor the numerator

Factor out the common factor from the numerator. In the expression \(\frac{3x+6}{3x}\), the numerator \(3x+6\) can be factored by taking out the common factor of 3: \(\frac{3(x+2)}{3x}\).

Step 3 - Simplify the expression

Cancel out the common factor of 3 from the numerator and the denominator: \(\frac{3(x+2)}{3x} = \frac{x+2}{x}\).

Step 4 - Verify the result

The simplified form of the rational expression is \(\frac{x+2}{x}\). Confirm that there are no further common factors to ensure it is in the lowest terms.

Key Concepts

Factoring

Factoring is an essential step in simplifying rational expressions. It involves breaking down a more complex expression into simpler factors that can be multiplied together. In the case of the expression \(\frac{3x+6}{3x}\), the numerator \(3x + 6\) can be factored by finding the greatest common factor (GCF). The GCF here is 3, making the factored form \(3(x + 2)\).

To find the GCF of a set of terms, look for the largest number or variable that is a factor of each term:

• Identify the numbers: For terms 3x and 6, the GCF of the numbers 3 and 6 is 3.
• Identify the variables: Since both terms share the variable x, include it if it appears in both terms.
• Combine these to get the GCF: In this case, it's 3.

Factoring makes it easy to simplify the expression, allowing us to cancel out common factors.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form so that the numerator and the denominator no longer share any common factors. Once the expression \(\frac{3(x+2)}{3x}\) is factored, we simplify it by canceling out the common factor of 3 in the numerator and denominator.

Here's how to do it:

• Identify the common factor: 3.
• Cancel the common factor: When you divide both the numerator and denominator by 3, you get \(\frac{x + 2}{x}\).

Simplifying fractions helps you see the underlying relationships in fractions and makes them easier to understand and work with in further calculations.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. These expressions follow the same rules as numerical fractions but often require factoring and simplification steps. The expression \(\frac{3x+6}{3x}\) simplifies as follows: After factoring, we get \(\frac{3(x+2)}{3x}\). Simplifying it by canceling out the common factor gives us the result \(\frac{x+2}{x}\).

To work with rational expressions, always:

• Factor the polynomials in both the numerator and the denominator.
• Cancel out any common factors to simplify.
• Check that the expression is in its lowest terms.

Understanding rational expressions makes algebraic operations like addition, subtraction, multiplication, and division more manageable.