Question

Classify the sum:

3r/5 + 6/2+r



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Answer 1

To classify the sum \( \frac{3r}{5} + \frac{6}{2+r} \), we can start by simplifying both terms.

1. The first term \( \frac{3r}{5} \) is already in its simplest form.
2. The second term \( \frac{6}{2+r} \) can also be expressed in a simpler way, but not much simplification is required unless we want a common denominator for easier addition.

### Common Denominator
To add these two fractions, we need a common denominator. The denominators are \( 5 \) and \( 2+r \). The common denominator would be \( 5(2+r) \).

Now we can rewrite each fraction:

- For \( \frac{3r}{5} \):
\[
\frac{3r}{5} \cdot \frac{2+r}{2+r} = \frac{3r(2+r)}{5(2+r)}
\]

- For \( \frac{6}{2+r} \):
\[
\frac{6}{2+r} \cdot \frac{5}{5} = \frac{30}{5(2+r)}
\]

### Add the fractions
Now we can add the two fractions together:
\[
\frac{3r(2+r) + 30}{5(2+r)}
\]

### Result
So the final expression is:
\[
\frac{3r(2+r) + 30}{5(2+r)}
\]

### Classification
This expression is a rational function because it is the ratio of two polynomials (the numerator and the denominator).

If you need further simplification of the numerator:
\[
3r(2+r) + 30 = 3r^2 + 6r + 30
\]

Thus, the full expression becomes:
\[
\frac{3r^2 + 6r + 30}{5(2+r)}
\]

To summarize: the expression \( \frac{3r}{5} + \frac{6}{2+r} \) is classified as a rational function.