Maths · Polynomial, rational, trigonometric, logarithmic and exponential functions
Find the Quotient and the Remainder when the first polynomial is divided by the
Find the Quotient and the Remainder when the first polynomial is divided by the second. \( \left(x^{4}-2 x^{3}+2 x^{2}+x+4\right) \) by \( \left(x^{2}+x+\right. \) 1)
- A. Quotient \( =x^{2}+3 x+4 \), Remainder \( =0 \)
- B. Quotient= \( x^{2}-3 x-4 \), Remainder \( =0 \)
- C. Quotient \( =-x^{2}-3 x+4 \), Remainder \( =0 \)
- D. Quotient \( =x^{2}-3 x+4 \), Remainder \( =0 \)
Step-by-step solution
Perform polynomial long division: Divide x^4 - 2x^3 + 2x^2 + x + 4 by x^2 + x + 1. First term: x^2 (since x^4/x^2 = x^2). Multiply divisor: x^4 + x^3 + x^2. Subtract: -3x^3 + x^2 + x + 4. Second term: -3x (since -3x^3/x^2 = -3x). Multiply: -3x^3 - 3x^2 - 3x. Subtract: 4x^2 + 4x + 4. Third term: 4 (since 4x^2/x^2 = 4). Multiply: 4x^2 + 4x + 4. Subtract: 0. Hence quotient = x^2 - 3x + 4, remainder = 0.