Final answer:
The remainder theorem for evaluating f(x) divided by x² isn't directly applicable here, but using the concept shows the remainder is -7, not 0. Hence, x² is not a factor of f(x). The answer is option d) Remainder: -7, x² is not a factor.
Step-by-step explanation:
To use the remainder theorem to find the remainder when f(x) is divided by x², we would ordinarily evaluate f(0) since x² = x - 0. However, this isn't actually applicable here since the remainder theorem typically applies to division by linear factors, not quadratic like x². But if we want to attempt this for a quadratic expression and judge by the result whether x² is a factor, the function f(x) = 2x¶ - 8x⁴x³ - 7 has a constant term -7, so the remainder when divided by x² would be -7.
The factor theorem states that if the remainder is 0 when f(x) is divided by a polynomial (x-r), then (x-r) is a factor of f(x). Since the remainder is not 0, x² is not a factor of f(x). The correct answer is: Remainder: -7, x² is not a factor.