At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
