Problem 16 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{6 a}{a-3}-\frac{18}{a-3} $$

Step-by-Step Solution

Answer

The simplified result is 6.

Step 1: Identify the Operation and Denominators

We need to determine whether we are adding or subtracting. In this problem, we are subtracting two rational expressions. Notice that both fractions, \( \frac{6a}{a-3} \) and \( \frac{18}{a-3} \), have the same denominator \( a-3 \).

Step 2: Combine the Fractions

Since the denominators are the same, simply combine the numerators by performing the indicated operation (subtraction) on them. This gives us: \[ \frac{6a - 18}{a-3} \].

Step 3: Simplify the Numerator

Factor the numerator \( 6a - 18 \). Notice that we can factor out a common term, which is 6. Thus, \( 6a - 18 = 6(a - 3) \).

Step 4: Simplify the Rational Expression

Since \( 6(a-3) \) is the numerator and \( a-3 \) is the denominator, we can simplify by canceling out the common factor \( a-3 \). This leaves us with \( 6 \).

Step 5: Present the Simplified Form

The simplified form of the original expression \( \frac{6a}{a-3} - \frac{18}{a-3} \) is \( 6 \).

Key Concepts

Simplifying Rational Expressions

Rational expressions are fractions where both numerator and denominator are polynomials. Simplifying them is crucial in making problems easier to solve, especially when combining or comparing fractions. To simplify a rational expression, follow these general steps:

• First, factor the numerator and the denominator. Look out for common polynomial factors that might cancel each other out.


• Next, cancel any common factors that appear in both the numerator and the denominator. Always ensure you only cancel factors that are indeed common to both.


• The result will be a simpler form of the original rational expression.

In our exercise, the expression \[\frac{6a - 18}{a-3}\]was simplified because the common factor \((a-3)\) in both the numerator and denominator could be canceled. Remember that the main goal is to reduce the complexity of the expression by eliminating unnecessary parts, like common factors, while keeping the value of the expression unchanged.

Subtraction of Rational Expressions

Subtracting rational expressions with the same denominator follows the same basic rules as subtracting regular fractions. If fractions have a common denominator, you can directly subtract their numerators. Here's a step-by-step approach:

• Ensure both rational expressions share the same denominator. If they do not, find a common denominator first.


• Subtract the second numerator from the first while keeping the common denominator unchanged.


• Simplify the result by factoring and reducing, if possible.

In the exercise given, the subtraction was straightforward because both rational expressions had the common denominator \(a-3\). This resulted in subtracting the numerators to form \[\frac{6a - 18}{a-3}\]. From there, we could then move to the simplification process, highlighting the difference in complexity when the denominators are identical versus when they're not.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that is often used in simplifying rational expressions. The process involves expressing a polynomial as the product of its factors, which can be constants or polynomials of lower degrees. Key points when factoring a polynomial include:

• Identify and factor out the greatest common factor (GCF) from all terms in the polynomial.


• For quadratic polynomials, use methods like factoring by grouping, or the special cases such as the difference of squares when applicable.


• Use trial and error or the factoring formula for higher-degree polynomials when necessary.

In the exercise, factoring was crucial in simplifying \[6a - 18 \] because it allowed us to recognize the common factor \(6\), leading to the expression \(6(a-3)\) which was pivotal in canceling out the duplicate \(a-3\) from the denominator. Understanding and efficiently factoring polynomials is a powerful tool in solving algebraic equations and simplifying expressions.