Answer :
To divide the polynomial [tex]\(12x^4 + 23x^3 - 9x^2 + 15x + 4\)[/tex] by [tex]\(3x - 1\)[/tex] using long division, follow these steps:
1. Set up the division:
Write the dividend [tex]\(12x^4 + 23x^3 - 9x^2 + 15x + 4\)[/tex] under the division bar and the divisor [tex]\(3x - 1\)[/tex] outside.
2. Divide the first term:
Divide the first term of the dividend [tex]\(12x^4\)[/tex] by the first term of the divisor [tex]\(3x\)[/tex]. This gives [tex]\(4x^3\)[/tex] as the first term of the quotient because [tex]\( \frac{12x^4}{3x} = 4x^3\)[/tex].
3. Multiply and subtract:
Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(3x - 1\)[/tex], giving [tex]\(12x^4 - 4x^3\)[/tex]. Subtract this result from the original dividend to get a new polynomial:
[tex]\((23x^3 - 9x^2) - (-4x^3) = 27x^3 - 9x^2\)[/tex].
4. Repeat the process:
- Divide the new first term [tex]\(27x^3\)[/tex] by [tex]\(3x\)[/tex], resulting in [tex]\(9x^2\)[/tex].
- Multiply [tex]\(9x^2\)[/tex] by [tex]\(3x - 1\)[/tex] to get [tex]\(27x^3 - 9x^2\)[/tex].
- Subtract this from [tex]\(27x^3 - 9x^2\)[/tex] to get [tex]\(0x^3 + 0x^2 + 15x\)[/tex].
5. Continue dividing:
- The next term is [tex]\(15x\)[/tex]. Divide [tex]\(15x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by [tex]\(3x - 1\)[/tex], resulting in [tex]\(15x - 5\)[/tex].
- Subtract to find the remainder: [tex]\(15x + 4 - (15x - 5) = 9\)[/tex].
6. Conclusion:
The quotient is [tex]\(4x^3 + 9x^2 + 5\)[/tex], and the remainder is [tex]\(9\)[/tex].
So, the division results in a quotient of [tex]\(4x^3 + 9x^2 + 5\)[/tex] and a remainder of [tex]\(9\)[/tex]. Thus, the answer can be expressed as:
[tex]\[ 12x^4 + 23x^3 - 9x^2 + 15x + 4 = (3x - 1)(4x^3 + 9x^2 + 5) + 9 \][/tex]
1. Set up the division:
Write the dividend [tex]\(12x^4 + 23x^3 - 9x^2 + 15x + 4\)[/tex] under the division bar and the divisor [tex]\(3x - 1\)[/tex] outside.
2. Divide the first term:
Divide the first term of the dividend [tex]\(12x^4\)[/tex] by the first term of the divisor [tex]\(3x\)[/tex]. This gives [tex]\(4x^3\)[/tex] as the first term of the quotient because [tex]\( \frac{12x^4}{3x} = 4x^3\)[/tex].
3. Multiply and subtract:
Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(3x - 1\)[/tex], giving [tex]\(12x^4 - 4x^3\)[/tex]. Subtract this result from the original dividend to get a new polynomial:
[tex]\((23x^3 - 9x^2) - (-4x^3) = 27x^3 - 9x^2\)[/tex].
4. Repeat the process:
- Divide the new first term [tex]\(27x^3\)[/tex] by [tex]\(3x\)[/tex], resulting in [tex]\(9x^2\)[/tex].
- Multiply [tex]\(9x^2\)[/tex] by [tex]\(3x - 1\)[/tex] to get [tex]\(27x^3 - 9x^2\)[/tex].
- Subtract this from [tex]\(27x^3 - 9x^2\)[/tex] to get [tex]\(0x^3 + 0x^2 + 15x\)[/tex].
5. Continue dividing:
- The next term is [tex]\(15x\)[/tex]. Divide [tex]\(15x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by [tex]\(3x - 1\)[/tex], resulting in [tex]\(15x - 5\)[/tex].
- Subtract to find the remainder: [tex]\(15x + 4 - (15x - 5) = 9\)[/tex].
6. Conclusion:
The quotient is [tex]\(4x^3 + 9x^2 + 5\)[/tex], and the remainder is [tex]\(9\)[/tex].
So, the division results in a quotient of [tex]\(4x^3 + 9x^2 + 5\)[/tex] and a remainder of [tex]\(9\)[/tex]. Thus, the answer can be expressed as:
[tex]\[ 12x^4 + 23x^3 - 9x^2 + 15x + 4 = (3x - 1)(4x^3 + 9x^2 + 5) + 9 \][/tex]
