To perform the division of the polynomial [tex]\(\frac{6x^3 + 11x^2 - 5x - 12}{3x + 4}\)[/tex], let's follow the process of polynomial long division step-by-step:
1. Setup the Division:
Set up the dividend [tex]\(6x^3 + 11x^2 - 5x - 12\)[/tex] and the divisor [tex]\(3x + 4\)[/tex].
2. First Division Step:
- Divide the leading term of the dividend ([tex]\(6x^3\)[/tex]) by the leading term of the divisor ([tex]\(3x\)[/tex]):
[tex]\[
\frac{6x^3}{3x} = 2x^2
\][/tex]
- Multiply the entire divisor by this result:
[tex]\[
2x^2 \cdot (3x + 4) = 6x^3 + 8x^2
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(6x^3 + 11x^2 - 5x - 12) - (6x^3 + 8x^2) = 3x^2 - 5x - 12
\][/tex]
3. Second Division Step:
- Divide the leading term of the new dividend ([tex]\(3x^2\)[/tex]) by the leading term of the divisor ([tex]\(3x\)[/tex]):
[tex]\[
\frac{3x^2}{3x} = x
\][/tex]
- Multiply the entire divisor by this result:
[tex]\[
x \cdot (3x + 4) = 3x^2 + 4x
\][/tex]
- Subtract this from the current polynomial:
[tex]\[
(3x^2 - 5x - 12) - (3x^2 + 4x) = -9x - 12
\][/tex]
4. Third Division Step:
- Divide the leading term of the new dividend ([tex]\(-9x\)[/tex]) by the leading term of the divisor ([tex]\(3x\)[/tex]):
[tex]\[
\frac{-9x}{3x} = -3
\][/tex]
- Multiply the entire divisor by this result:
[tex]\[
-3 \cdot (3x + 4) = -9x - 12
\][/tex]
- Subtract this from the current polynomial:
[tex]\[
(-9x - 12) - (-9x - 12) = 0
\][/tex]
5. Conclusion:
- After performing these steps, we find that the quotient is [tex]\(2x^2 + x - 3\)[/tex] and the remainder is [tex]\(0\)[/tex].
So the result of the division [tex]\( \frac{6x^3 + 11x^2 - 5x - 12}{3x + 4} \)[/tex] is:
[tex]\[
2x^2 + x - 3 \quad \text{with a remainder of } 0
\][/tex]
Thus, we express our final answer:
[tex]\[
\frac{6x^3 + 11x^2 - 5x - 12}{3x + 4} = 2x^2 + x - 3
\][/tex]
