The value of `(3x ^(3) + 9x ^(2) + 27x)div 3x` is

Options

A. `x ^(2) + 9 + 27 x`
B. `3x ^(3) + 3x ^(2) + 27x`
C. `3x ^(3) + 9x ^(2) + 9`
D. `x ^(2) + 3x +9`

Answer

To solve the expression \((3x^3 + 9x^2 + 27x) \div 3x\), we will follow the steps of polynomial long division.

### Step-by-Step Solution:

1. **Write the expression**: We start with the expression:
\[ \frac{3x^3 + 9x^2 + 27x}{3x} \]

2. **Divide the first term**: Divide the first term of the numerator \(3x^3\) by the denominator \(3x\):
\[ \frac{3x^3}{3x} = x^2 \]
So, the first term of the quotient is \(x^2\).

3. **Multiply and subtract**: Now, multiply \(x^2\) by the entire denominator \(3x\):
\[ x^2 \cdot 3x = 3x^3 \]
Subtract this from the original polynomial:
\[ (3x^3 + 9x^2 + 27x) - 3x^3 = 9x^2 + 27x \]

4. **Repeat the process**: Now, take the new expression \(9x^2 + 27x\) and divide the first term by \(3x\):
\[ \frac{9x^2}{3x} = 3x \]
So, the next term of the quotient is \(3x\).

5. **Multiply and subtract again**: Multiply \(3x\) by \(3x\):
\[ 3x \cdot 3x = 9x^2 \]
Subtract this from \(9x^2 + 27x\):
\[ (9x^2 + 27x) - 9x^2 = 27x \]

6. **Final division**: Now, divide \(27x\) by \(3x\):
\[ \frac{27x}{3x} = 9 \]
So, the last term of the quotient is \(9\).

7. **Final multiplication and subtraction**: Multiply \(9\) by \(3x\):
\[ 9 \cdot 3x = 27x \]
Subtract this from \(27x\):
\[ 27x - 27x = 0 \]

8. **Write the final answer**: Since there are no remaining terms, the quotient is:
\[ x^2 + 3x + 9 \]

### Final Result:
The value of \((3x^3 + 9x^2 + 27x) \div 3x\) is:
\[ \boxed{x^2 + 3x + 9} \]