To solve the polynomial division [tex]\(\frac{6x^3 + 11x^2 - 5x - 12}{3x + 4}\)[/tex], we will perform a step-by-step polynomial long division.
### Step 1: Setup
Set up the division as you would with numerical long division. The numerator is [tex]\(6x^3 + 11x^2 - 5x - 12\)[/tex] and the denominator is [tex]\(3x + 4\)[/tex].
### Step 2: First Division
Divide the leading term of the numerator, [tex]\(6x^3\)[/tex], by the leading term of the denominator, [tex]\(3x\)[/tex]:
[tex]\[
\frac{6x^3}{3x} = 2x^2
\][/tex]
Write [tex]\(2x^2\)[/tex] as the first term of the quotient.
### Step 3: Multiply and Subtract
Multiply [tex]\(2x^2\)[/tex] by the entire denominator [tex]\(3x + 4\)[/tex]:
[tex]\[
2x^2 \cdot (3x + 4) = 6x^3 + 8x^2
\][/tex]
Subtract this result from the original numerator:
[tex]\[
(6x^3 + 11x^2 - 5x - 12) - (6x^3 + 8x^2) = 3x^2 - 5x - 12
\][/tex]
### Step 4: Second Division
Divide the new leading term of the numerator, [tex]\(3x^2\)[/tex], by the leading term of the denominator, [tex]\(3x\)[/tex]:
[tex]\[
\frac{3x^2}{3x} = x
\][/tex]
Write [tex]\(x\)[/tex] as the next term of the quotient.
### Step 5: Multiply and Subtract
Multiply [tex]\(x\)[/tex] by the entire denominator [tex]\(3x + 4\)[/tex]:
[tex]\[
x \cdot (3x + 4) = 3x^2 + 4x
\][/tex]
Subtract this result from the current numerator:
[tex]\[
(3x^2 - 5x - 12) - (3x^2 + 4x) = -9x - 12
\][/tex]
### Step 6: Third Division
Divide the new leading term of the numerator, [tex]\(-9x\)[/tex], by the leading term of the denominator, [tex]\(3x\)[/tex]:
[tex]\[
\frac{-9x}{3x} = -3
\][/tex]
Write [tex]\(-3\)[/tex] as the next term of the quotient.
### Step 7: Multiply and Subtract
Multiply [tex]\(-3\)[/tex] by the entire denominator [tex]\(3x + 4\)[/tex]:
[tex]\[
-3 \cdot (3x + 4) = -9x - 12
\][/tex]
Subtract this result from the current numerator:
[tex]\[
(-9x - 12) - (-9x - 12) = 0
\][/tex]
### Conclusion
The division process gives us:
[tex]\[
\frac{6x^3 + 11x^2 - 5x - 12}{3x + 4} = 2x^2 + x - 3
\][/tex]
The quotient is [tex]\(2x^2 + x - 3\)[/tex] and the remainder is 0.
So, the simplified version of the polynomial division is indeed:
[tex]\[
2x^2 + x - 3
\][/tex]
