Final answer:
After applying the Remainder Theorem, it is concluded that the binomial (n + 8) is indeed a factor of the polynomial (n⁴ + 9n³ + 14n² + 50n + 9), as the substitution n = -8 into the polynomial yields a result of zero.
Step-by-step explanation:
To determine if the binomial (n + 8) is a factor of the polynomial (n⁴ + 9n³ + 14n² + 50n + 9), we can use the Remainder Theorem. The Remainder Theorem states that if a binomial (x - c) is a factor of a polynomial f(x), then the value of f(c) will be zero. In this case, we need to evaluate the polynomial at n = -8.
Let's substitute n = -8 into the polynomial:
(n⁴ + 9n³ + 14n² + 50n + 9) evaluated at n = -8 becomes:
((-8)⁴ + 9(-8)³ + 14(-8)² + 50(-8) + 9) = (4096 - 4608 + 896 - 400 + 9).
Calculating the value, we get:
(4096 - 4608 + 896 - 400 + 9) = 0.
Since the result is zero, we conclude that (n + 8) is indeed a factor of (n⁴ + 9n³ + 14n² + 50n + 9).
Regarding part b) of the question, this is simply a contradictory statement to part a), so if we have established that (n + 8) is a factor based on the above calculation, then the statement that it is not a factor is incorrect.