Final answer:
To find the polynomial g(x), we used the relationship that the dividend equals the divisor times the quotient plus the remainder. Upon substituting the quotient and remainder and simplifying, we determined that g(x) is the second degree polynomial x² - x + 1.
Step-by-step explanation:
To find the polynomial g(x), we use the relationship given by the division of polynomials, which states that the dividend is equal to the divisor times the quotient plus the remainder. From our problem, we have:
x³ - 3x² + x + 2 = g(x) · (x - 2) + (-2x + 4)
We substitute the given quotient (x - 2) and remainder (-2x + 4) into the equation and simplify:
x³ - 3x² + x + 2 = g(x) · (x - 2) - 2x + 4
To find g(x), we distribute g(x) into the quotient and solve for g(x):
g(x) · (x - 2) = x³ - 3x² + x + 2 - (-2x + 4)
g(x) · (x - 2) = x³ - 3x² + 3x - 2
Since the degree of g(x) must be one less than the degree of the dividend when the remainder is not zero, g(x) is a second degree polynomial. We match the coefficients on both sides of the equation to find:
g(x) = x² - x + 1