Answer:
Option B. x - 2
Explanation:
1. We proceed to consider the dividend x^4 −2x^3 − x + 2 as the first "Remainder".
2. Divide the principal term of the remainder, (x^4), by the principal term of the denominator, (x^3):
x^4 / x^3 = x, the result x is the first term of the quotient.
3. We take the term we just obtained and multiply it by the denominator:
x⋅ (x^3 − 1) = x^4 − x
4. We subtract from the numerator the result of the previous multiplication, giving rise to the next remainder.
-> New remainder: x^4 − 2x^3 – x + 2 - (x^4 − x) = x^4 − 2x^3 – x + 2 – x^4 + x = −2x^3 + 2
New remainder: −2x^3 + 2
5. Divide the principal term of the remainder, (−2x^3), by the principal term of the denominator, (x^3):
−2x^3 / x^3 = -2, the result −2 is the next quotient term.
So far the quotient is: x − 2
6. We take the term we just obtained and multiply it by the denominator:
−2⋅ (x^3 − 1) = −2x^3 + 2
7. We subtract from the numerator the result of the previous multiplication, giving rise to the next remainder.
-> New remainder: −2x^3 + 2 - (−2x^3 + 2) = −2x^3 + 2 + 2x^3 - 2 = 0
New remainder: 0
8. There is no remainder, which indicates the end of the division process.
9. The final result of this polynomial division is the quotient:
x – 2