The quotient obtained when ` x^4 + 11 x^3-44 x^2 + 76 x+ 48 ` is divided by `x^2 -2x+1` is

Options

A. `x^2 -13 x +5`
B. `x^2 +13 x-19`
C. `x^2 -13x +19`
D. `x^2 +13x +25`

Answer

To find the quotient when \( x^4 + 11x^3 - 44x^2 + 76x + 48 \) is divided by \( x^2 - 2x + 1 \), we will perform polynomial long division.

### Step-by-Step Solution:

1. **Set up the division**: We are dividing \( x^4 + 11x^3 - 44x^2 + 76x + 48 \) by \( x^2 - 2x + 1 \).

2. **Divide the leading terms**: Divide the leading term of the dividend \( x^4 \) by the leading term of the divisor \( x^2 \):
\[ \frac{x^4}{x^2} = x^2 \]
This is the first term of the quotient.

3. **Multiply and subtract**: Multiply \( x^2 \) by the entire divisor \( x^2 - 2x + 1 \):
\[ x^2(x^2 - 2x + 1) = x^4 - 2x^3 + x^2 \]
Now subtract this from the original polynomial:
\[ (x^4 + 11x^3 - 44x^2 + 76x + 48) - (x^4 - 2x^3 + x^2) = (11x^3 + 2x^3) + (-44x^2 - x^2) + 76x + 48 \]
Simplifying this gives:
\[ 13x^3 - 45x^2 + 76x + 48 \]

4. **Repeat the process**: Now, divide the leading term \( 13x^3 \) by \( x^2 \):
\[ \frac{13x^3}{x^2} = 13x \]
This is the next term of the quotient.

5. **Multiply and subtract again**: Multiply \( 13x \) by the divisor:
\[ 13x(x^2 - 2x + 1) = 13x^3 - 26x^2 + 13x \]
Now subtract this from the current polynomial:
\[ (13x^3 - 45x^2 + 76x + 48) - (13x^3 - 26x^2 + 13x) = (-45x^2 + 26x^2) + (76x - 13x) + 48 \]
Simplifying gives:
\[ -19x^2 + 63x + 48 \]

6. **Continue the process**: Divide the leading term \( -19x^2 \) by \( x^2 \):
\[ \frac{-19x^2}{x^2} = -19 \]
This is the next term of the quotient.

7. **Final multiplication and subtraction**: Multiply \( -19 \) by the divisor:
\[ -19(x^2 - 2x + 1) = -19x^2 + 38x - 19 \]
Subtract this from the current polynomial:
\[ (-19x^2 + 63x + 48) - (-19x^2 + 38x - 19) = (63x - 38x) + (48 + 19) \]
Simplifying gives:
\[ 25x + 67 \]

8. **Conclusion**: The quotient obtained from the division is:
\[ x^2 + 13x - 19 \]
with a remainder of \( 25x + 67 \).

### Final Answer:
The quotient is \( x^2 + 13x - 19 \).