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Answer :
To solve the polynomial division problem [tex]\((3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9) \div (3x - 4)\)[/tex], we will use polynomial long division. Let's go through the steps:
1. Setup the Division:
- Dividend: [tex]\(3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9\)[/tex]
- Divisor: [tex]\(3x - 4\)[/tex]
2. Perform Long Division:
- Step 1: Divide the leading term of the dividend [tex]\(3x^5\)[/tex] by the leading term of the divisor [tex]\(3x\)[/tex] to get [tex]\(x^4\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(3x - 4\)[/tex] by [tex]\(x^4\)[/tex] (which is [tex]\(3x^5 - 4x^4\)[/tex]) and subtract it from the dividend.
- Step 3: The new polynomial is [tex]\(-27x^4 + 54x^3 - 30x^2 - 4x + 9\)[/tex].
Repeat similar steps:
- Divide [tex]\(-27x^4\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-9x^3\)[/tex].
- Multiply the divisor by [tex]\(-9x^3\)[/tex] to get [tex]\(-27x^4 + 36x^3\)[/tex] and subtract.
- New polynomial: [tex]\(18x^3 - 30x^2 - 4x + 9\)[/tex].
- Divide [tex]\(18x^3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(6x^2\)[/tex].
- Multiply the divisor by [tex]\(6x^2\)[/tex] to get [tex]\(18x^3 - 24x^2\)[/tex] and subtract.
- New polynomial: [tex]\(-6x^2 - 4x + 9\)[/tex].
- Divide [tex]\(-6x^2\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-2x\)[/tex].
- Multiply the divisor by [tex]\(-2x\)[/tex] to get [tex]\(-6x^2 + 8x\)[/tex] and subtract.
- New polynomial: [tex]\(-12x + 9\)[/tex].
- Divide [tex]\(-12x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-4\)[/tex].
- Multiply the divisor by [tex]\(-4\)[/tex] to get [tex]\(-12x + 16\)[/tex] and subtract.
- Final remainder: [tex]\(-7\)[/tex].
3. Construct the Result:
- The quotient is [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4\)[/tex].
- The remainder is [tex]\(-7\)[/tex].
So, the answer is option B: [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4, \text{ R } -7\)[/tex].
1. Setup the Division:
- Dividend: [tex]\(3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9\)[/tex]
- Divisor: [tex]\(3x - 4\)[/tex]
2. Perform Long Division:
- Step 1: Divide the leading term of the dividend [tex]\(3x^5\)[/tex] by the leading term of the divisor [tex]\(3x\)[/tex] to get [tex]\(x^4\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(3x - 4\)[/tex] by [tex]\(x^4\)[/tex] (which is [tex]\(3x^5 - 4x^4\)[/tex]) and subtract it from the dividend.
- Step 3: The new polynomial is [tex]\(-27x^4 + 54x^3 - 30x^2 - 4x + 9\)[/tex].
Repeat similar steps:
- Divide [tex]\(-27x^4\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-9x^3\)[/tex].
- Multiply the divisor by [tex]\(-9x^3\)[/tex] to get [tex]\(-27x^4 + 36x^3\)[/tex] and subtract.
- New polynomial: [tex]\(18x^3 - 30x^2 - 4x + 9\)[/tex].
- Divide [tex]\(18x^3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(6x^2\)[/tex].
- Multiply the divisor by [tex]\(6x^2\)[/tex] to get [tex]\(18x^3 - 24x^2\)[/tex] and subtract.
- New polynomial: [tex]\(-6x^2 - 4x + 9\)[/tex].
- Divide [tex]\(-6x^2\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-2x\)[/tex].
- Multiply the divisor by [tex]\(-2x\)[/tex] to get [tex]\(-6x^2 + 8x\)[/tex] and subtract.
- New polynomial: [tex]\(-12x + 9\)[/tex].
- Divide [tex]\(-12x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-4\)[/tex].
- Multiply the divisor by [tex]\(-4\)[/tex] to get [tex]\(-12x + 16\)[/tex] and subtract.
- Final remainder: [tex]\(-7\)[/tex].
3. Construct the Result:
- The quotient is [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4\)[/tex].
- The remainder is [tex]\(-7\)[/tex].
So, the answer is option B: [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4, \text{ R } -7\)[/tex].
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