To simplify the expression [tex]\(\frac{p(x)}{q(x)} + \frac{r(x)}{s(x)}\)[/tex], we can follow these steps:
1. Identify the fractions and their denominators:
- The first fraction is [tex]\(\frac{p(x)}{q(x)}\)[/tex].
- The second fraction is [tex]\(\frac{r(x)}{s(x)}\)[/tex].
2. Find a common denominator:
- The denominators of the two fractions are [tex]\(q(x)\)[/tex] and [tex]\(s(x)\)[/tex].
- The common denominator is the product of these two denominators, which is [tex]\(q(x) \cdot s(x)\)[/tex].
3. Express each fraction with the common denominator:
- To convert [tex]\(\frac{p(x)}{q(x)}\)[/tex] to the common denominator, multiply the numerator and denominator by [tex]\(s(x)\)[/tex]:
[tex]\[
\frac{p(x)}{q(x)} \cdot \frac{s(x)}{s(x)} = \frac{p(x) \cdot s(x)}{q(x) \cdot s(x)}
\][/tex]
- To convert [tex]\(\frac{r(x)}{s(x)}\)[/tex] to the common denominator, multiply the numerator and denominator by [tex]\(q(x)\)[/tex]:
[tex]\[
\frac{r(x)}{s(x)} \cdot \frac{q(x)}{q(x)} = \frac{r(x) \cdot q(x)}{s(x) \cdot q(x)}
\][/tex]
4. Add the fractions:
- With a common denominator, the sum of the fractions becomes:
[tex]\[
\frac{p(x) \cdot s(x)}{q(x) \cdot s(x)} + \frac{r(x) \cdot q(x)}{s(x) \cdot q(x)} = \frac{p(x) \cdot s(x) + r(x) \cdot q(x)}{q(x) \cdot s(x)}
\][/tex]
5. Simplify the result:
- The simplified form of the expression is the combined numerator over the common denominator:
[tex]\[
\frac{p(x) \cdot s(x) + r(x) \cdot q(x)}{q(x) \cdot s(x)}
\][/tex]
This is the simplified form of the given expression:
[tex]\[
\frac{p(x)}{q(x)} + \frac{r(x)}{s(x)} = \frac{p(x) \cdot s(x) + r(x) \cdot q(x)}{q(x) \cdot s(x)}
\][/tex]
