Divide the polynomial `9x^(3) -3x^(2)+15x-3` by (3x-1) and find its quotient and remainder.

Answer

To divide the polynomial \( 9x^3 - 3x^2 + 15x - 3 \) by \( 3x - 1 \), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
Write the dividend \( 9x^3 - 3x^2 + 15x - 3 \) and the divisor \( 3x - 1 \).

### Step 2: Divide the leading term
Divide the leading term of the dividend \( 9x^3 \) by the leading term of the divisor \( 3x \):
\[ \frac{9x^3}{3x} = 3x^2 \]
This is the first term of the quotient.

### Step 3: Multiply and subtract
Multiply the entire divisor \( 3x - 1 \) by \( 3x^2 \):
\[ 3x^2(3x - 1) = 9x^3 - 3x^2 \]
Now, subtract this from the original polynomial:
\[ (9x^3 - 3x^2 + 15x - 3) - (9x^3 - 3x^2) = 0 + 15x - 3 = 15x - 3 \]

### Step 4: Repeat the process
Now, take the new polynomial \( 15x - 3 \) and divide its leading term \( 15x \) by the leading term of the divisor \( 3x \):
\[ \frac{15x}{3x} = 5 \]
This is the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply the entire divisor \( 3x - 1 \) by \( 5 \):
\[ 5(3x - 1) = 15x - 5 \]
Now, subtract this from the current polynomial:
\[ (15x - 3) - (15x - 5) = 0 + 2 = 2 \]

### Step 6: Write the final result
Now, we have completed the division. The quotient is \( 3x^2 + 5 \) and the remainder is \( 2 \).

Thus, the final result is:
- **Quotient**: \( 3x^2 + 5 \)
- **Remainder**: \( 2 \)

---