Problem 46 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{3 x+7}{3 x+10}$$

Step-by-Step Solution

Answer

The rational expression \(\frac{3x+7}{3x+10}\) is already in its simplest form and cannot be further simplified.

Step 1: Factor both the numerator and the denominator

The first process in simplifying any rational expression is to factor the numerator and the denominator. This gives us: Numerator: \(3x+7\) - There are no common factors other than 1 in the terms of the numerator, therefore, it can't be factored.Denominator: \(3x+10\) - There are also no common factors other than 1 in the terms of the denominator, thus, it also can't be factored.

Step 2: Cancel common factors

After factoring, we look for any common factors between the numerator and the denominator. However, for this given rational expression, since the numerator and the denominator can't be factored and have no common factors other than 1, there are no terms that can be cancelled out. This implies that the original expression is already in its simplest form.

Step 3: Check final expression

Our rational expression is still \(\frac{3x+7}{3x+10}\), as we found no common factors to simplify the expression.