To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that complete the sum:
[tex]\[
\frac{3}{x} + \frac{5}{x^2} = \frac{a x + b}{x^2},
\][/tex]
we need to follow a step-by-step process of combining the fractions on the left side of the equation. Here's the detailed process:
1. Write down the given fractions and find a common denominator:
The common denominator for [tex]\(\frac{3}{x}\)[/tex] and [tex]\(\frac{5}{x^2}\)[/tex] is [tex]\(x^2\)[/tex].
2. Express each fraction with the common denominator:
[tex]\[
\frac{3}{x} = \frac{3 \cdot x}{x^2} = \frac{3x}{x^2},
\][/tex]
and
[tex]\[
\frac{5}{x^2} = \frac{5}{x^2}.
\][/tex]
3. Combine the fractions into a single fraction:
Add the numerators together while keeping the common denominator:
[tex]\[
\frac{3x}{x^2} + \frac{5}{x^2} = \frac{3x + 5}{x^2}.
\][/tex]
4. Identify the numerator and the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
By comparing the combined fraction [tex]\(\frac{3x + 5}{x^2}\)[/tex] with the form [tex]\(\frac{a x + b}{x^2}\)[/tex], we can directly see that:
[tex]\[
3x + 5 = a x + b.
\][/tex]
5. Match the coefficients:
From the equation above, we match the coefficients respectively. The coefficient of [tex]\(x\)[/tex] gives us the value of [tex]\(a\)[/tex] and the constant term gives us the value of [tex]\(b\)[/tex]:
- The coefficient of [tex]\(x\)[/tex] is 3, so [tex]\(a = 3\)[/tex].
- The constant term is 5, so [tex]\(b = 5\)[/tex].
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
\begin{array}{l}
a = 3, \\
b = 5.
\end{array}
\][/tex]
So the completed sum is:
[tex]\[
\frac{3}{x} + \frac{5}{x^2} = \frac{3x + 5}{x^2}.
\][/tex]
