Questions: The partial fraction decomposition of (x^2+10)/((x-3)(x^2+4)) can be written in the form of (f(x))/(x-3)+(g(x))/(x^2+4). What type of function will the numerators f(x) and g(x) be? f(x) is select an answer and g(x) is select an answer

$The partial fraction decomposition of (x^2+10)/((x-3)(x^2+4)) can be written in the form of (f(x))/(x-3)+(g(x))/(x^2+4). What type of function will the numerators f(x) and g(x) be? f(x) is select an answer and g(x) is select an answer$

Transcript text: The partial fraction decomposition of $\frac{x^{2}+10}{(x-3)\left(x^{2}+4\right)}$ can be written in the form of $\frac{f(x)}{x-3}+\frac{g(x)}{x^{2}+4}$. What type of function will the numerators $f(x)$ and $g(x)$ be? $f(x)$ is select an answer and $g(x)$ is Select an answer

Solution

Solution Steps

Step 1: Understand the Problem

We are given a rational function $\frac{x^{2}+10}{(x-3)\left(x^{2}+4\right)}$ and need to express it in the form of partial fraction decomposition: $\frac{f(x)}{x-3}+\frac{g(x)}{x^{2}+4}$. We need to determine the types of functions $f(x)$ and $g(x)$ will be.

Step 2: Set Up the Partial Fraction Decomposition

The partial fraction decomposition of $\frac{x^{2}+10}{(x-3)(x^{2}+4)}$ can be expressed as:

\[ \frac{A}{x-3} + \frac{Bx + C}{x^{2}+4} \]

where $A$, $B$, and $C$ are constants to be determined.

Step 3: Determine the Types of Functions

$f(x)$: The numerator of the term $\frac{A}{x-3}$ is a constant, so $f(x)$ is a constant function.
$g(x)$: The numerator of the term $\frac{Bx + C}{x^{2}+4}$ is a linear polynomial, so $g(x)$ is a linear function.