Find the quotient and the remainder (if any), when :
`6x^(2) + x - 15` is divided by 3x + 5. In each case, verify your answer.

Answer

To find the quotient and the remainder when dividing \(6x^2 + x - 15\) by \(3x + 5\), we will use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We want to divide \(6x^2 + x - 15\) by \(3x + 5\). We can write this as:

\[ \frac{6x^2 + x - 15}{3x + 5} \]

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(6x^2\) by the leading term of the divisor \(3x\):

\[ \frac{6x^2}{3x} = 2x \]

This \(2x\) will be the first term of our quotient.

### Step 3: Multiply and subtract
Now, multiply \(2x\) by the entire divisor \(3x + 5\):

\[ 2x \cdot (3x + 5) = 6x^2 + 10x \]

Now, subtract this result from the original polynomial:

\[ (6x^2 + x - 15) - (6x^2 + 10x) = x - 10x - 15 = -9x - 15 \]

### Step 4: Repeat the process
Now, we need to divide \(-9x\) by \(3x\):

\[ \frac{-9x}{3x} = -3 \]

This \(-3\) will be the next term of our quotient.

### Step 5: Multiply and subtract again
Now, multiply \(-3\) by the divisor \(3x + 5\):

\[ -3 \cdot (3x + 5) = -9x - 15 \]

Subtract this from the current polynomial:

\[ (-9x - 15) - (-9x - 15) = 0 \]

### Step 6: Write the final result
Now that we have no remainder, we can conclude that:

- The quotient is \(2x - 3\)
- The remainder is \(0\)

Thus, the final answer is:

\[ \text{Quotient} = 2x - 3, \quad \text{Remainder} = 0 \]

### Verification
To verify, we can multiply the quotient by the divisor and add the remainder:

\[ (2x - 3)(3x + 5) + 0 = 6x^2 + 10x - 9x - 15 = 6x^2 + x - 15 \]

This matches the original polynomial, confirming our division was performed correctly.