Find the quotient when `3x^(2)-7x-6` is divided by (x-3).

Answer

To find the quotient when \(3x^2 - 7x - 6\) is divided by \(x - 3\), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We will set up the long division with \(3x^2 - 7x - 6\) as the dividend and \(x - 3\) as the divisor.

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(3x^2\) by the leading term of the divisor \(x\):
\[ \frac{3x^2}{x} = 3x \]
This means \(3x\) is the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \(3x\) by the entire divisor \(x - 3\):
\[ 3x \cdot (x - 3) = 3x^2 - 9x \]
Next, subtract this result from the original polynomial:
\[ (3x^2 - 7x - 6) - (3x^2 - 9x) = -7x + 9x - 6 = 2x - 6 \]

### Step 4: Repeat the process
Now, we will repeat the process with the new polynomial \(2x - 6\). Divide the leading term \(2x\) by the leading term \(x\):
\[ \frac{2x}{x} = 2 \]
So, \(2\) is the next term of the quotient.

### Step 5: Multiply and subtract again
Now, multiply \(2\) by the entire divisor \(x - 3\):
\[ 2 \cdot (x - 3) = 2x - 6 \]
Subtract this from \(2x - 6\):
\[ (2x - 6) - (2x - 6) = 0 \]

### Conclusion
Since the remainder is \(0\), we have completed the division. The quotient is:
\[ 3x + 2 \]

### Final Answer
The quotient when \(3x^2 - 7x - 6\) is divided by \(x - 3\) is \(3x + 2\).

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