Let's perform the polynomial long division of [tex]\( \frac{x^4 - x^2 + 12}{x^3 - x^2} \)[/tex] and then carry out the partial fraction decomposition of the remainder term.
### Step 1: Polynomial Long Division
1. Divide the leading term:
[tex]\( \frac{x^4}{x^3} = x \)[/tex]
2. Multiply and subtract:
[tex]\[
(x^3 - x^2) \cdot x = x^4 - x^3
\][/tex]
Subtract this from the original dividend:
[tex]\[
(x^4 - x^2) - (x^4 - x^3) = x^3 - x^2
\][/tex]
3. Divide the next leading term:
[tex]\( \frac{x^3 - x^2}{x^3 - x^2} = 1 \)[/tex]
4. Multiply and subtract:
[tex]\[
(x^3 - x^2) \cdot 1 = x^3 - x^2
\][/tex]
Subtract this from the result:
[tex]\[
(x^3 - x^2 + 12) - (x^3 - x^2) = 12
\][/tex]
By performing the above, we see that:
[tex]\[
\frac{x^4 - x^2 + 12}{x^3 - x^2} = x + 1
\][/tex]
with a remainder of 12.
### Step 2: Express the Remainder over the Divisor and Perform Partial Fraction Decomposition
We re-write the division result:
[tex]\[
\frac{x^4 - x^2 + 12}{x^3 - x^2} = x + 1 + \frac{12}{x^3 - x^2}
\][/tex]
Next, we focus on the term [tex]\(\frac{12}{x^3 - x^2}\)[/tex]. We can factorize the denominator:
[tex]\[
x^3 - x^2 = x^2 (x - 1)
\][/tex]
Thus, we aim to decompose:
[tex]\[
\frac{12}{x^2 (x - 1)}
\][/tex]
We use partial fraction decomposition for [tex]\(\frac{12}{x^2 (x - 1)}\)[/tex]:
[tex]\[
\frac{12}{x^2 (x - 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1}
\][/tex]
To find [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], we equate and solve:
[tex]\[
12 = A x (x - 1) + B (x - 1) + C x^2
\][/tex]
Matching coefficients:
- For [tex]\(x^2\)[/tex]: [tex]\(Ax - A + B = 0 \Rightarrow A = C\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(Ax = 0\)[/tex] implies [tex]\(A = 0\)[/tex]
- For constants: [tex]\(B = 12\)[/tex]
Thus:
[tex]\[
A = 12, B = -12, C = -12
\][/tex]
Substituting back:
[tex]\[
\frac{12}{x^2 (x - 1)} = \frac{-12}{x} + \frac{-12}{x^2} + \frac{12}{x - 1}
\][/tex]
So, our final decomposition is:
[tex]\[
x + 1 + \frac{12}{x - 1} - \frac{12}{x} - \frac{12}{x^2}
\][/tex]
Thus, the final answer to the problem is:
[tex]\[
\frac{x^4 - x^2 + 12}{x^3 - x^2} = x + 1 + \frac{12}{x - 1} - \frac{12}{x} - \frac{12}{x^2}
\][/tex]
with the quotient as [tex]\(x + 1\)[/tex], remainder as [tex]\(12\)[/tex], and the partial fraction decomposition given.
