Sure! Let's use polynomial long division to find the quotient and remainder for the given division:
[tex]\[ \frac{12x^3 - 11x^2 + 9x + 18}{4x + 3} \][/tex]
Here's the step-by-step process:
1. Set up the division:
Write the dividend ( [tex]\(12x^3 - 11x^2 + 9x + 18\)[/tex] ) inside the division symbol and the divisor ( [tex]\(4x + 3\)[/tex] ) outside.
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[ \frac{12x^3}{4x} = 3x^2 \][/tex]
This is the first term of the quotient.
3. Multiply the entire divisor by this term:
[tex]\[ 3x^2 \cdot (4x + 3) = 12x^3 + 9x^2 \][/tex]
4. Subtract the result from the dividend:
[tex]\[ (12x^3 - 11x^2 + 9x + 18) - (12x^3 + 9x^2) = -20x^2 + 9x + 18 \][/tex]
5. Repeat the process with the new polynomial:
- Divide the first term:
[tex]\[ \frac{-20x^2}{4x} = -5x \][/tex]
This is the next term of the quotient.
- Multiply the entire divisor:
[tex]\[ -5x \cdot (4x + 3) = -20x^2 - 15x \][/tex]
- Subtract from the current polynomial:
[tex]\[ (-20x^2 + 9x + 18) - (-20x^2 - 15x) = 24x + 18 \][/tex]
6. Repeat the process one more time:
- Divide the first term:
[tex]\[ \frac{24x}{4x} = 6 \][/tex]
This is the final term of the quotient.
- Multiply the entire divisor:
[tex]\[ 6 \cdot (4x + 3) = 24x + 18 \][/tex]
- Subtract from the current polynomial:
[tex]\[ (24x + 18) - (24x + 18) = 0 \][/tex]
Since we are left with a remainder of 0, our final quotient is [tex]\( 3x^2 - 5x + 6 \)[/tex].
7. Combine the result:
The quotient is [tex]\[ 3x^2 - 5x + 6 \][/tex] and the remainder is [tex]\[ 0 \][/tex].
Therefore, the result of the division is:
[tex]\[ 3x^2 - 5x + 6 + \frac{0}{4x + 3} \][/tex]
or simply:
[tex]\[ 3x^2 - 5x + 6 \][/tex]
