Problem 34 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{2 m-2 n}{4 n-4 m}$$

Step-by-Step Solution

Answer

-\(\frac{1}{2}\)

Step 1: Factor the numerator

Identify any common factors in the numerator. The numerator is \(2m - 2n\). Notice that 2 is a common factor, so factor it out: \[2m - 2n = 2(m - n)\].

Step 2: Factor the denominator

Identify any common factors in the denominator. The denominator is \(4n - 4m\). Notice that 4 is a common factor, so factor it out: \[4n - 4m = 4(n - m)\].

Step 3: Simplify the rational expression

Rewrite the rational expression with the factored forms: \[\frac{2(m - n)}{4(n - m)}\]. Notice that \(n - m\) is the negative of \(m - n\). Thus, \(4(n - m)\) can be rewritten as \(-4(m - n)\). Now the expression is: \[\frac{2(m - n)}{-4(m - n)}\].

Step 4: Cancel common factors

Cancel out the common factor \((m - n)\) from the numerator and the denominator: \[\frac{2(m - n)}{-4(m - n)} = \frac{2}{-4} = -\frac{1}{2}\].

Key Concepts

factoring polynomials

Understanding how to factor polynomials is crucial when simplifying rational expressions. Factoring means breaking down a polynomial into simpler 'factors' that, when multiplied together, give you the original polynomial. For example, consider the polynomial in the numerator of our problem, \(2m - 2n\). Notice that each term contains the number 2 as a factor. Hence, we factor it out:

• \(2m - 2n\) becomes \(2(m - n)\).

Similarly, we apply the same technique to the denominator \(4n - 4m\), factoring out the common factor 4:

• \(4n - 4m\) becomes \(4(n - m)\).

numerator and denominator

The terms numerator and denominator are fundamental when dealing with fractions and rational expressions.

• The numerator is the top part of a fraction.
• The denominator is the bottom part of a fraction.

In our example, the original fraction's numerator is \(2m - 2n\) and the denominator is \(4n - 4m\). By factoring, we get:

• Numerator: \(2(m - n)\)
• Denominator: \(4(n - m)\)

To rewrite the rational expression clearly:

• \(\frac{2(m - n)}{4(n - m)}\)

cancelling common factors

To simplify a rational expression, you can cancel out common factors from the numerator and denominator. In our example, we start with the fraction:

• \(\frac{2(m - n)}{4(n - m)}\)

Notice that \(n - m\) is the negative of \(m - n\). Thus, we can rewrite \(4(n - m)\) as \(-4(m - n)\). Now the expression looks like this:

• \(\frac{2(m - n)}{-4(m - n)}\)

At this step, \(m - n\) is a common factor in both the numerator and denominator and can be cancelled out:

• \(\frac{2(m - n)}{-4(m - n)} = \frac{2}{-4} = -\frac{1}{2}\)

Through this step-by-step method, the rational expression simplifies to \(-\frac{1}{2}\).