What is the quotient when `10a^(2)+3a-27` is divided by `2a-3`

Options

A. `5a-9`
B. `(-5a-9)`
C. `(-5a+9)`
D. `5a+9`

Answer

To find the quotient when \(10a^2 + 3a - 27\) is divided by \(2a - 3\), we will use polynomial long division. Here’s the step-by-step solution:

### Step 1: Set up the division
We write \(10a^2 + 3a - 27\) (the dividend) under the long division symbol and \(2a - 3\) (the divisor) outside.

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(10a^2\) by the leading term of the divisor \(2a\):
\[ \frac{10a^2}{2a} = 5a \]
This means \(5a\) is the first term of our quotient.

### Step 3: Multiply and subtract
Now, multiply \(5a\) by the entire divisor \(2a - 3\):
\[ 5a \cdot (2a - 3) = 10a^2 - 15a \]
Next, subtract this result from the original polynomial:
\[ (10a^2 + 3a - 27) - (10a^2 - 15a) = 3a + 15a - 27 = 18a - 27 \]

### Step 4: Repeat the process
Now, we take the new polynomial \(18a - 27\) and repeat the process. Divide the leading term \(18a\) by the leading term of the divisor \(2a\):
\[ \frac{18a}{2a} = 9 \]
So, \(9\) is the next term of our quotient.

### Step 5: Multiply and subtract again
Now, multiply \(9\) by the entire divisor \(2a - 3\):
\[ 9 \cdot (2a - 3) = 18a - 27 \]
Subtract this from \(18a - 27\):
\[ (18a - 27) - (18a - 27) = 0 \]

### Conclusion
Since the remainder is \(0\), we have completed the division. The quotient is:
\[ \text{Quotient} = 5a + 9 \]

### Final Answer
The quotient when \(10a^2 + 3a - 27\) is divided by \(2a - 3\) is:
\[ \boxed{5a + 9} \]