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Sagot :
To determine the quotient [tex]\(Q(x)\)[/tex] and the remainder [tex]\(R(x)\)[/tex] when the polynomial [tex]\(P(x) = x^3 - 3x^2 + 4x - 2\)[/tex] is divided by [tex]\(D(x) = x - 2\)[/tex], we can manually perform polynomial long division.
### Step-by-Step Polynomial Division:
1. Setup the Division:
[tex]\[ \text{Divide } x^3 - 3x^2 + 4x - 2 \text{ by } x - 2. \][/tex]
2. First Division:
- Divide the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{x^3}{x} = x^2 \][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^2 \cdot (x - 2) = x^3 - 2x^2 \][/tex]
- Subtract this result from the original polynomial:
[tex]\[ (x^3 - 3x^2 + 4x - 2) - (x^3 - 2x^2) = -x^2 + 4x - 2 \][/tex]
3. Second Division:
- Divide the leading term of the new dividend [tex]\(-x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{-x^2}{x} = -x \][/tex]
- Multiply [tex]\(-x\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ -x \cdot (x - 2) = -x^2 + 2x \][/tex]
- Subtract this result from the new dividend:
[tex]\[ (-x^2 + 4x - 2) - (-x^2 + 2x) = 2x - 2 \][/tex]
4. Third Division:
- Divide the leading term of the new dividend [tex]\(2x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{x} = 2 \][/tex]
- Multiply [tex]\(2\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 2 \cdot (x - 2) = 2x - 4 \][/tex]
- Subtract this result from the new dividend:
[tex]\[ (2x - 2) - (2x - 4) = -2 + 4 = 2 \][/tex]
Hence, the quotient [tex]\(Q(x)\)[/tex] is:
[tex]\[ Q(x) = x^2 - x + 2 \][/tex]
And the remainder [tex]\(R(x)\)[/tex] is:
[tex]\[ R(x) = 2 \][/tex]
### Verification (Using Substitution or Synthetic Division):
We substitute [tex]\(x = 2\)[/tex] into the original polynomial to verify the remainder:
[tex]\[ P(2) = 2^3 - 3 \cdot 2^2 + 4 \cdot 2 - 2 = 8 - 12 + 8 - 2 = 2 \][/tex]
Thus, the remainder when [tex]\(x = 2\)[/tex] is indeed 2.
### Conclusion:
With the found quotient and remainder, the correct answer is:
b) [tex]\(Q(x) = x^2 - x + 2 \)[/tex] and [tex]\(R(x) = 2\)[/tex]
### Step-by-Step Polynomial Division:
1. Setup the Division:
[tex]\[ \text{Divide } x^3 - 3x^2 + 4x - 2 \text{ by } x - 2. \][/tex]
2. First Division:
- Divide the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{x^3}{x} = x^2 \][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^2 \cdot (x - 2) = x^3 - 2x^2 \][/tex]
- Subtract this result from the original polynomial:
[tex]\[ (x^3 - 3x^2 + 4x - 2) - (x^3 - 2x^2) = -x^2 + 4x - 2 \][/tex]
3. Second Division:
- Divide the leading term of the new dividend [tex]\(-x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{-x^2}{x} = -x \][/tex]
- Multiply [tex]\(-x\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ -x \cdot (x - 2) = -x^2 + 2x \][/tex]
- Subtract this result from the new dividend:
[tex]\[ (-x^2 + 4x - 2) - (-x^2 + 2x) = 2x - 2 \][/tex]
4. Third Division:
- Divide the leading term of the new dividend [tex]\(2x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{x} = 2 \][/tex]
- Multiply [tex]\(2\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 2 \cdot (x - 2) = 2x - 4 \][/tex]
- Subtract this result from the new dividend:
[tex]\[ (2x - 2) - (2x - 4) = -2 + 4 = 2 \][/tex]
Hence, the quotient [tex]\(Q(x)\)[/tex] is:
[tex]\[ Q(x) = x^2 - x + 2 \][/tex]
And the remainder [tex]\(R(x)\)[/tex] is:
[tex]\[ R(x) = 2 \][/tex]
### Verification (Using Substitution or Synthetic Division):
We substitute [tex]\(x = 2\)[/tex] into the original polynomial to verify the remainder:
[tex]\[ P(2) = 2^3 - 3 \cdot 2^2 + 4 \cdot 2 - 2 = 8 - 12 + 8 - 2 = 2 \][/tex]
Thus, the remainder when [tex]\(x = 2\)[/tex] is indeed 2.
### Conclusion:
With the found quotient and remainder, the correct answer is:
b) [tex]\(Q(x) = x^2 - x + 2 \)[/tex] and [tex]\(R(x) = 2\)[/tex]
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