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Answer :
Certainly! Let's find the quotient and remainder when dividing the polynomial [tex]\(x^5 - x^3 + x - 5\)[/tex] by [tex]\(x - 2\)[/tex].
### Step-by-Step Process
1. Set up the division:
- Dividend: [tex]\(x^5 - x^3 + x - 5\)[/tex]
- Divisor: [tex]\(x - 2\)[/tex]
2. Perform polynomial division:
- Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(x^5\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4\)[/tex].
- Multiply entire divisor by this result:
- Multiply [tex]\(x^4\)[/tex] by [tex]\(x - 2\)[/tex] to get [tex]\(x^5 - 2x^4\)[/tex].
- Subtract this from the dividend:
- This creates a new polynomial: [tex]\((- x^3 + x - 5 - (-2x^4)) = 2x^4 - x^3 + x - 5\)[/tex].
- Repeat the process:
- Divide the first term [tex]\(2x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(2x^3\)[/tex].
- Multiply [tex]\(2x^3\)[/tex] by [tex]\(x - 2\)[/tex] to get [tex]\(2x^4 - 4x^3\)[/tex].
- Subtract:
[tex]\( (-x^3 + x - 5 - (-4x^3)) = -x^3 + x - 5 + 4x^3 = 3x^3 + x - 5 \)[/tex].
- Continue this pattern:
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(3x^2\)[/tex].
- Multiply: [tex]\(3x^2(x - 2) = 3x^3 - 6x^2\)[/tex].
- Subtract: [tex]\((x - 5 - (-6x^2)) = 6x^2 + x - 5\)[/tex].
- Next, divide: [tex]\(6x^2/x = 6x\)[/tex].
- Multiply: [tex]\(6x(x - 2) = 6x^2 - 12x\)[/tex].
- Subtract: [tex]\((x - 5 + 12x) = 13x - 5\)[/tex].
- Lastly, divide: [tex]\(13x/x = 13\)[/tex].
- Multiply: [tex]\(13(x - 2) = 13x - 26\)[/tex].
- Subtract: [tex]\((-5 + 26) = 21\)[/tex].
3. Find the final remainder:
- The final expression after the subtraction indicates that the remainder is 21.
4. Write the result:
- The quotient is [tex]\(x^4 + 2x^3 + 3x^2 + 6x + 13\)[/tex].
- The remainder is 21.
Hence, the quotient and remainder of the division are:
Quotient: [tex]\(x^4 + 2x^3 + 3x^2 + 6x + 13\)[/tex]
Remainder: 21
Therefore, the correct answer is option d, which matches these results:
- [tex]\(x^4 + 2x^3 + 3x^2 + 6x + 13\)[/tex]
- Remainder: 21
### Step-by-Step Process
1. Set up the division:
- Dividend: [tex]\(x^5 - x^3 + x - 5\)[/tex]
- Divisor: [tex]\(x - 2\)[/tex]
2. Perform polynomial division:
- Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(x^5\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4\)[/tex].
- Multiply entire divisor by this result:
- Multiply [tex]\(x^4\)[/tex] by [tex]\(x - 2\)[/tex] to get [tex]\(x^5 - 2x^4\)[/tex].
- Subtract this from the dividend:
- This creates a new polynomial: [tex]\((- x^3 + x - 5 - (-2x^4)) = 2x^4 - x^3 + x - 5\)[/tex].
- Repeat the process:
- Divide the first term [tex]\(2x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(2x^3\)[/tex].
- Multiply [tex]\(2x^3\)[/tex] by [tex]\(x - 2\)[/tex] to get [tex]\(2x^4 - 4x^3\)[/tex].
- Subtract:
[tex]\( (-x^3 + x - 5 - (-4x^3)) = -x^3 + x - 5 + 4x^3 = 3x^3 + x - 5 \)[/tex].
- Continue this pattern:
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(3x^2\)[/tex].
- Multiply: [tex]\(3x^2(x - 2) = 3x^3 - 6x^2\)[/tex].
- Subtract: [tex]\((x - 5 - (-6x^2)) = 6x^2 + x - 5\)[/tex].
- Next, divide: [tex]\(6x^2/x = 6x\)[/tex].
- Multiply: [tex]\(6x(x - 2) = 6x^2 - 12x\)[/tex].
- Subtract: [tex]\((x - 5 + 12x) = 13x - 5\)[/tex].
- Lastly, divide: [tex]\(13x/x = 13\)[/tex].
- Multiply: [tex]\(13(x - 2) = 13x - 26\)[/tex].
- Subtract: [tex]\((-5 + 26) = 21\)[/tex].
3. Find the final remainder:
- The final expression after the subtraction indicates that the remainder is 21.
4. Write the result:
- The quotient is [tex]\(x^4 + 2x^3 + 3x^2 + 6x + 13\)[/tex].
- The remainder is 21.
Hence, the quotient and remainder of the division are:
Quotient: [tex]\(x^4 + 2x^3 + 3x^2 + 6x + 13\)[/tex]
Remainder: 21
Therefore, the correct answer is option d, which matches these results:
- [tex]\(x^4 + 2x^3 + 3x^2 + 6x + 13\)[/tex]
- Remainder: 21
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