To determine the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the given expression
[tex]\[
\frac{3}{x} + \frac{5}{x^2} = \frac{ax + b}{x^2},
\][/tex]
we need to rewrite the left-hand side of the equation so that it has a common denominator.
The first fraction is
[tex]\[
\frac{3}{x},
\][/tex]
and the second fraction is
[tex]\[
\frac{5}{x^2}.
\][/tex]
To add these fractions, we need a common denominator. The least common denominator here is [tex]\( x^2 \)[/tex]. We rewrite each fraction with this common denominator:
[tex]\[
\frac{3}{x} = \frac{3 \cdot x}{x \cdot x} = \frac{3x}{x^2},
\][/tex]
[tex]\[
\frac{5}{x^2} = \frac{5}{x^2}.
\][/tex]
Now, we can add these fractions:
[tex]\[
\frac{3x}{x^2} + \frac{5}{x^2} = \frac{3x + 5}{x^2}.
\][/tex]
We want to express this as
[tex]\[
\frac{ax + b}{x^2}.
\][/tex]
Comparing the numerators, we find that
[tex]\[
3x + 5 = ax + b.
\][/tex]
From this equation, we can see that:
1. The coefficient of [tex]\( x \)[/tex] on the left side is 3, so [tex]\( a = 3 \)[/tex].
2. The constant term on the left side is 5, so [tex]\( b = 5 \)[/tex].
Therefore, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[
a = 3
\][/tex]
[tex]\[
b = 5
\][/tex]
