To rewrite the given fraction [tex]\(\frac{-2 x^2 - 14 x - 49}{x^3 - 7 x^2}\)[/tex] as a sum of partial fractions, we follow these steps:
1. Factorize the denominator:
The denominator [tex]\(x^3 - 7 x^2\)[/tex] can be factorized as:
[tex]\[
x^3 - 7x^2 = x^2(x - 7)
\][/tex]
2. Set up the form for partial fractions:
Since the denominator [tex]\(x^2(x - 7)\)[/tex] has the factors [tex]\(x^2\)[/tex] and [tex]\((x - 7)\)[/tex], we can decompose the fraction into the form:
[tex]\[
\frac{-2 x^2 - 14 x - 49}{x^2 (x - 7)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 7}
\][/tex]
3. Combine the partial fractions into a single fraction:
To combine the fractions, we bring them over a common denominator:
[tex]\[
\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 7} = \frac{A(x - 7) + B(x - 7) + Cx^2}{x^2(x - 7)}
\][/tex]
4. Equate numerators of the fractions:
Now, we obtain:
[tex]\[
-2 x^2 - 14 x - 49 = A(x - 7) + B(x - 7) + Cx^2
\][/tex]
5. Simplify and collect like terms:
Combine and collect the terms involving [tex]\(x\)[/tex]:
[tex]\[
-2 x^2 - 14 x - 49 = (A+C)x^2 + (-7A+B)x - 7B
\][/tex]
6. Matching coefficients:
Compare coefficients of the like terms [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant terms on both sides of the equation:
[tex]\[
\begin{cases}
A + C = -2 \\
-7A + B = -14 \\
-7B = -49 \\
\end{cases}
\][/tex]
7. Solve the system of equations:
- From the equation [tex]\(-7B = -49\)[/tex], we get:
[tex]\[
B = 7
\][/tex]
- Substitute [tex]\(B = 7\)[/tex] in the equation [tex]\(-7A + 7 = -14\)[/tex]:
[tex]\[
-7A + 7 = -14 \\
-7A = -21 \\
A = 3
\][/tex]
- Substitute [tex]\(A = 3\)[/tex] in the equation [tex]\(A + C = -2\)[/tex]:
[tex]\[
3 + C = -2 \\
C = -5
\][/tex]
8. Write out the partial fractions:
By substituting [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] back into the partial fractions form:
[tex]\[
\frac{-2 x^2 - 14 x - 49}{x^2 (x - 7)} = \frac{3}{x} + \frac{7}{x^2} + \frac{-5}{x - 7}
\][/tex]
Simplify the fractions:
[tex]\[
\frac{-2 x^2 - 14 x - 49}{x^2 (x - 7)} = \frac{3}{x} + \frac{7}{x^2} - \frac{5}{x - 7}
\][/tex]
Therefore, the partial fraction decomposition of the given expression is:
[tex]\[
\frac{-2 x^2 - 14 x - 49}{x^3 - 7 x^2} = \frac{3}{x} + \frac{7}{x^2} - \frac{5}{x - 7}
\][/tex]
