When `x^(3) - 3x^(2) + 3x + 5` is divided by `x^(2)` - x + 1, the quotient and remainder are

Options

A. `x+2 ,7`
B. x-2 , -7
C. x- 2 , 7
D. x + 2 , -7

Answer

To solve the problem of dividing the polynomial \( x^3 - 3x^2 + 3x + 5 \) by \( x^2 - x + 1 \), we will perform polynomial long division. Let's go through the steps:

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

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

### Step 3: Multiply and subtract
Now, multiply the entire divisor \( x^2 - x + 1 \) by \( x \):
\[ x(x^2 - x + 1) = x^3 - x^2 + x \]
Subtract this from the original polynomial:
\[ (x^3 - 3x^2 + 3x + 5) - (x^3 - x^2 + x) = (-3x^2 + x^2) + (3x - x) + 5 = -2x^2 + 2x + 5 \]

### Step 4: Repeat the process
Now, we need to divide the new polynomial \( -2x^2 + 2x + 5 \) by \( x^2 - x + 1 \).

Divide the leading term \( -2x^2 \) by \( x^2 \):
\[ \frac{-2x^2}{x^2} = -2 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply the divisor by \(-2\):
\[ -2(x^2 - x + 1) = -2x^2 + 2x - 2 \]
Subtract this from the polynomial:
\[ (-2x^2 + 2x + 5) - (-2x^2 + 2x - 2) = 5 + 2 = 7 \]

### Conclusion
At this point, we cannot divide further since the degree of the remainder (7) is less than the degree of the divisor (\(x^2 - x + 1\)).

Thus, the quotient is \( x - 2 \) and the remainder is \( 7 \).

### Final Answer
- Quotient: \( x - 2 \)
- Remainder: \( 7 \)