Divide :
`6x^(3) - 13x^(2) - 13x + 30` by `2x^(2) - x - 6`

Answer

To divide the polynomial \(6x^3 - 13x^2 - 13x + 30\) by \(2x^2 - x - 6\), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We will divide \(6x^3 - 13x^2 - 13x + 30\) by \(2x^2 - x - 6\).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(6x^3\) by the leading term of the divisor \(2x^2\):
\[ \frac{6x^3}{2x^2} = 3x \]

### Step 3: Multiply and subtract
Now, multiply the entire divisor \(2x^2 - x - 6\) by \(3x\):
\[ 3x(2x^2 - x - 6) = 6x^3 - 3x^2 - 18x \]
Now, subtract this from the original polynomial:
\[ (6x^3 - 13x^2 - 13x + 30) - (6x^3 - 3x^2 - 18x) = (-13x^2 + 3x^2) + (-13x + 18x) + 30 \]
This simplifies to:
\[ -10x^2 + 5x + 30 \]

### Step 4: Repeat the process
Now we need to divide \(-10x^2 + 5x + 30\) by \(2x^2 - x - 6\). Divide the leading term:
\[ \frac{-10x^2}{2x^2} = -5 \]
Now multiply the entire divisor by \(-5\):
\[ -5(2x^2 - x - 6) = -10x^2 + 5x + 30 \]
Subtract this from \(-10x^2 + 5x + 30\):
\[ (-10x^2 + 5x + 30) - (-10x^2 + 5x + 30) = 0 \]

### Step 5: Conclusion
Since the remainder is \(0\), we conclude that:
\[ \frac{6x^3 - 13x^2 - 13x + 30}{2x^2 - x - 6} = 3x - 5 \]

### Final Answer:
The result of the division is:
\[ 3x - 5 \]