To perform the partial fraction decomposition of the given rational expression:

$\frac{x^2 + 7}{x^3 + 5x^2},$

the denominator $x^3 + 5x^2$ can be factored as:

$x^3 + 5x^2 = x^2(x + 5).$

Thus, the expression can be decomposed into:

$\frac{x^2 + 7}{x^3 + 5x^2} = \frac{f(x)}{x} + \frac{g(x)}{x^2} + \frac{h(x)}{x + 5}.$

Here, $f(x), g(x), h(x)$ are constants, since the degree of the numerators in the partial fractions must be less than the degree of their respective denominators.

Step 1: Write the equation

Start with:

$\frac{x^2 + 7}{x^3 + 5x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 5}.$

Step 2: Combine the fractions

Multiply through by the denominator $x^3 + 5x^2 = x^2(x + 5)$ to eliminate the fractions:

$x^2 + 7 = A x(x + 5) + B(x + 5) + Cx^2.$

Step 3: Expand the terms

Expand the right-hand side:

$x^2 + 7 = A(x^2 + 5x) + B(x + 5) + Cx^2.$

Simplify:

$x^2 + 7 = Ax^2 + 5Ax + Bx + 5B + Cx^2.$

Group like terms:

$x^2 + 7 = (A + C)x^2 + (5A + B)x + 5B.$

Step 4: Equate coefficients

Equate the coefficients of $x^2$, $x$, and the constant term on both sides:

1. Coefficient of $x^2$: $A + C = 1$,
2. Coefficient of $x$: $5A + B = 0$,
3. Constant term: $5B = 7$.

Step 5: Solve the system of equations

1. From $5B = 7$, solve for $B$: $B = \frac{7}{5}.$

2. Substitute $B = \frac{7}{5}$ into $5A + B = 0$: $5A + \frac{7}{5} = 0 \implies 5A = -\frac{7}{5} \implies A = -\frac{7}{25}.$

3. Use $A + C = 1$ to solve for $C$: $-\frac{7}{25} + C = 1 \implies C = 1 + \frac{7}{25} = \frac{25}{25} + \frac{7}{25} = \frac{32}{25}.$

Final Values

The coefficients are:

$f(x) = A = -\frac{7}{25}, \quad g(x) = B = \frac{7}{5}, \quad h(x) = C = \frac{32}{25}.$

Final Decomposition

The partial fraction decomposition is:

$\frac{x^2 + 7}{x^3 + 5x^2} = \frac{-\frac{7}{25}}{x} + \frac{\frac{7}{5}}{x^2} + \frac{\frac{32}{25}}{x + 5}.$

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Related Questions:

1. How is partial fraction decomposition used to integrate rational functions?
2. Can partial fraction decomposition be applied to improper fractions? If yes, how?
3. What are the common applications of partial fractions in engineering or physics?
4. How do you handle repeated roots in the denominator for partial fraction decomposition?
5. What is the role of linear vs quadratic factors in partial fraction decomposition?

Tip:

Always check if the rational function is improper. If it is, perform polynomial long division before proceeding with partial fractions.