Answer :
To divide the polynomial [tex]\((12x^4 + 23x^3 - 9x^2 + 15x + 4)\)[/tex] by [tex]\((3x - 1)\)[/tex], we can perform polynomial long division. Here's how we can do it step-by-step:
1. Setup the Division:
- Write [tex]\((12x^4 + 23x^3 - 9x^2 + 15x + 4)\)[/tex] as the dividend under the division symbol.
- Write [tex]\((3x - 1)\)[/tex] as the divisor outside the division symbol.
2. First Division:
- Determine how many times the first term of the divisor [tex]\(3x\)[/tex] goes into the first term of the dividend [tex]\(12x^4\)[/tex].
- Divide [tex]\(12x^4\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(4x^3\)[/tex].
- Multiply the entire divisor [tex]\(3x - 1\)[/tex] by [tex]\(4x^3\)[/tex] to get [tex]\(12x^4 - 4x^3\)[/tex].
- Subtract [tex]\((12x^4 - 4x^3)\)[/tex] from the first two terms of the dividend, resulting in a new polynomial: [tex]\(0x^4 + 27x^3\)[/tex].
3. Second Division:
- Bring down the next term to get [tex]\(27x^3 - 9x^2\)[/tex].
- Divide [tex]\(27x^3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(9x^2\)[/tex].
- Multiply the divisor [tex]\(3x - 1\)[/tex] by [tex]\(9x^2\)[/tex] to get [tex]\(27x^3 - 9x^2\)[/tex].
- Subtract [tex]\((27x^3 - 9x^2)\)[/tex] to get a new polynomial: [tex]\(0x^3 + 0x^2 + 15x\)[/tex].
4. Third Division:
- Bring down the next term to get [tex]\(15x\)[/tex].
- Divide [tex]\(15x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(5x\)[/tex].
- Multiply the divisor [tex]\(3x - 1\)[/tex] by [tex]\(5x\)[/tex] to get [tex]\(15x - 5\)[/tex].
- Subtract [tex]\((15x - 5)\)[/tex] to get a remainder of [tex]\(9\)[/tex].
5. Result:
- The quotient from this division is [tex]\(4x^3 + 9x^2 + 5\)[/tex].
- The remainder is [tex]\(9\)[/tex].
Thus, the result of the division [tex]\((12x^4 + 23x^3 - 9x^2 + 15x + 4) \div (3x - 1)\)[/tex] is:
Quotient: [tex]\(4x^3 + 9x^2 + 5\)[/tex]
Remainder: [tex]\(9\)[/tex]
1. Setup the Division:
- Write [tex]\((12x^4 + 23x^3 - 9x^2 + 15x + 4)\)[/tex] as the dividend under the division symbol.
- Write [tex]\((3x - 1)\)[/tex] as the divisor outside the division symbol.
2. First Division:
- Determine how many times the first term of the divisor [tex]\(3x\)[/tex] goes into the first term of the dividend [tex]\(12x^4\)[/tex].
- Divide [tex]\(12x^4\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(4x^3\)[/tex].
- Multiply the entire divisor [tex]\(3x - 1\)[/tex] by [tex]\(4x^3\)[/tex] to get [tex]\(12x^4 - 4x^3\)[/tex].
- Subtract [tex]\((12x^4 - 4x^3)\)[/tex] from the first two terms of the dividend, resulting in a new polynomial: [tex]\(0x^4 + 27x^3\)[/tex].
3. Second Division:
- Bring down the next term to get [tex]\(27x^3 - 9x^2\)[/tex].
- Divide [tex]\(27x^3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(9x^2\)[/tex].
- Multiply the divisor [tex]\(3x - 1\)[/tex] by [tex]\(9x^2\)[/tex] to get [tex]\(27x^3 - 9x^2\)[/tex].
- Subtract [tex]\((27x^3 - 9x^2)\)[/tex] to get a new polynomial: [tex]\(0x^3 + 0x^2 + 15x\)[/tex].
4. Third Division:
- Bring down the next term to get [tex]\(15x\)[/tex].
- Divide [tex]\(15x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(5x\)[/tex].
- Multiply the divisor [tex]\(3x - 1\)[/tex] by [tex]\(5x\)[/tex] to get [tex]\(15x - 5\)[/tex].
- Subtract [tex]\((15x - 5)\)[/tex] to get a remainder of [tex]\(9\)[/tex].
5. Result:
- The quotient from this division is [tex]\(4x^3 + 9x^2 + 5\)[/tex].
- The remainder is [tex]\(9\)[/tex].
Thus, the result of the division [tex]\((12x^4 + 23x^3 - 9x^2 + 15x + 4) \div (3x - 1)\)[/tex] is:
Quotient: [tex]\(4x^3 + 9x^2 + 5\)[/tex]
Remainder: [tex]\(9\)[/tex]
