Answer :
Sure, let's tackle this problem step by step.
### Step 1: Polynomial Division
We are asked to divide [tex]\(x^4 - x^2 + 9\)[/tex] by [tex]\(x^3 - x^2\)[/tex].
#### Setup for Long Division
1. Numerator (Dividend): [tex]\(x^4 - x^2 + 9\)[/tex]
2. Denominator (Divisor): [tex]\(x^3 - x^2\)[/tex]
We will perform polynomial long division.
#### Division Steps:
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
2. Multiply [tex]\(x\)[/tex] by the entire denominator:
[tex]\[ x \cdot (x^3 - x^2) = x^4 - x^3 \][/tex]
3. Subtract this product from the original numerator:
[tex]\[ (x^4 - x^2 + 9) - (x^4 - x^3) = x^3 - x^2 + 9 \][/tex]
4. Repeat the steps with the new polynomial [tex]\(x^3 - x^2 + 9\)[/tex]:
[tex]\[ \frac{x^3}{x^3} = 1 \][/tex]
So, the next term of the quotient is [tex]\(1\)[/tex].
5. Multiply [tex]\(1\)[/tex] by the entire denominator:
[tex]\[ 1 \cdot (x^3 - x^2) = x^3 - x^2 \][/tex]
6. Subtract this product from [tex]\(x^3 - x^2 + 9\)[/tex]:
[tex]\[ (x^3 - x^2 + 9) - (x^3 - x^2) = 9 \][/tex]
#### Quotient and Remainder:
At this stage, we can't divide further because the degree of the remainder (9) is less than the degree of the divisor (which is 3).
So, the division gives:
- Quotient: [tex]\(x + 1\)[/tex]
- Remainder: [tex]\(9\)[/tex]
Thus, we have:
[tex]\[ \frac{x^4 - x^2 + 9}{x^3 - x^2} = x + 1 + \frac{9}{x^3 - x^2} \][/tex]
### Step 2: Partial Fraction Decomposition of the Remainder
We need to decompose [tex]\(\frac{9}{x^3 - x^2}\)[/tex] into partial fractions.
First, factor the denominator:
[tex]\[ x^3 - x^2 = x^2 (x - 1) \][/tex]
So, we can write:
[tex]\[ \frac{9}{x^3 - x^2} = \frac{9}{x^2 (x - 1)} \][/tex]
To decompose it into partial fractions, we assume:
[tex]\[ \frac{9}{x^2 (x - 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 1} \][/tex]
We need to find constants [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]. Using the method of equating coefficients or plugging in convenient values, we find:
1. Multiply through by the common denominator [tex]\(x^2 (x - 1)\)[/tex]:
[tex]\[ 9 = A x (x - 1) + B (x - 1) + C x^2 \][/tex]
2. Plug in [tex]\(x = 0\)[/tex]:
[tex]\[ 9 = B(-1) \quad \implies \quad B = -9 \][/tex]
3. Plug in [tex]\(x = 1\)[/tex]:
[tex]\[ 9 = C (1)^2 \quad \implies \quad C = 9 \][/tex]
4. To determine [tex]\(A\)[/tex], compare coefficients for [tex]\(x^2\)[/tex]:
- The right-hand side has [tex]\((A + C)x^2\)[/tex].
- The left-hand side has no [tex]\(x^2\)[/tex] term.
Therefore:
[tex]\[ A + 9 = 0 \quad \implies \quad A = -9 \][/tex]
Thus, the partial fraction decomposition is:
[tex]\[ \frac{9}{x^2 (x - 1)} = \frac{-9}{x} + \frac{-9}{x^2} + \frac{9}{x - 1} \][/tex]
### Final Expression:
Combining everything, the solution to the problem is:
[tex]\[ \frac{x^4 - x^2 + 9}{x^3 - x^2} = x + 1 + \left(\frac{9}{x - 1} - \frac{9}{x} - \frac{9}{x^2}\right) \][/tex]
So, breaking it down:
- Quotient: [tex]\(x + 1\)[/tex]
- Remainder after Partial Fraction Decomposition: [tex]\(\frac{9}{x - 1} - \frac{9}{x} - \frac{9}{x^2}\)[/tex]
Thus, the final answer is:
[tex]\[ x + 1 + \frac{9}{x - 1} - \frac{9}{x} - \frac{9}{x^2} \][/tex]
### Step 1: Polynomial Division
We are asked to divide [tex]\(x^4 - x^2 + 9\)[/tex] by [tex]\(x^3 - x^2\)[/tex].
#### Setup for Long Division
1. Numerator (Dividend): [tex]\(x^4 - x^2 + 9\)[/tex]
2. Denominator (Divisor): [tex]\(x^3 - x^2\)[/tex]
We will perform polynomial long division.
#### Division Steps:
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
2. Multiply [tex]\(x\)[/tex] by the entire denominator:
[tex]\[ x \cdot (x^3 - x^2) = x^4 - x^3 \][/tex]
3. Subtract this product from the original numerator:
[tex]\[ (x^4 - x^2 + 9) - (x^4 - x^3) = x^3 - x^2 + 9 \][/tex]
4. Repeat the steps with the new polynomial [tex]\(x^3 - x^2 + 9\)[/tex]:
[tex]\[ \frac{x^3}{x^3} = 1 \][/tex]
So, the next term of the quotient is [tex]\(1\)[/tex].
5. Multiply [tex]\(1\)[/tex] by the entire denominator:
[tex]\[ 1 \cdot (x^3 - x^2) = x^3 - x^2 \][/tex]
6. Subtract this product from [tex]\(x^3 - x^2 + 9\)[/tex]:
[tex]\[ (x^3 - x^2 + 9) - (x^3 - x^2) = 9 \][/tex]
#### Quotient and Remainder:
At this stage, we can't divide further because the degree of the remainder (9) is less than the degree of the divisor (which is 3).
So, the division gives:
- Quotient: [tex]\(x + 1\)[/tex]
- Remainder: [tex]\(9\)[/tex]
Thus, we have:
[tex]\[ \frac{x^4 - x^2 + 9}{x^3 - x^2} = x + 1 + \frac{9}{x^3 - x^2} \][/tex]
### Step 2: Partial Fraction Decomposition of the Remainder
We need to decompose [tex]\(\frac{9}{x^3 - x^2}\)[/tex] into partial fractions.
First, factor the denominator:
[tex]\[ x^3 - x^2 = x^2 (x - 1) \][/tex]
So, we can write:
[tex]\[ \frac{9}{x^3 - x^2} = \frac{9}{x^2 (x - 1)} \][/tex]
To decompose it into partial fractions, we assume:
[tex]\[ \frac{9}{x^2 (x - 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 1} \][/tex]
We need to find constants [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]. Using the method of equating coefficients or plugging in convenient values, we find:
1. Multiply through by the common denominator [tex]\(x^2 (x - 1)\)[/tex]:
[tex]\[ 9 = A x (x - 1) + B (x - 1) + C x^2 \][/tex]
2. Plug in [tex]\(x = 0\)[/tex]:
[tex]\[ 9 = B(-1) \quad \implies \quad B = -9 \][/tex]
3. Plug in [tex]\(x = 1\)[/tex]:
[tex]\[ 9 = C (1)^2 \quad \implies \quad C = 9 \][/tex]
4. To determine [tex]\(A\)[/tex], compare coefficients for [tex]\(x^2\)[/tex]:
- The right-hand side has [tex]\((A + C)x^2\)[/tex].
- The left-hand side has no [tex]\(x^2\)[/tex] term.
Therefore:
[tex]\[ A + 9 = 0 \quad \implies \quad A = -9 \][/tex]
Thus, the partial fraction decomposition is:
[tex]\[ \frac{9}{x^2 (x - 1)} = \frac{-9}{x} + \frac{-9}{x^2} + \frac{9}{x - 1} \][/tex]
### Final Expression:
Combining everything, the solution to the problem is:
[tex]\[ \frac{x^4 - x^2 + 9}{x^3 - x^2} = x + 1 + \left(\frac{9}{x - 1} - \frac{9}{x} - \frac{9}{x^2}\right) \][/tex]
So, breaking it down:
- Quotient: [tex]\(x + 1\)[/tex]
- Remainder after Partial Fraction Decomposition: [tex]\(\frac{9}{x - 1} - \frac{9}{x} - \frac{9}{x^2}\)[/tex]
Thus, the final answer is:
[tex]\[ x + 1 + \frac{9}{x - 1} - \frac{9}{x} - \frac{9}{x^2} \][/tex]
