Final answer:
To determine if d(x) is a factor of f(x), we can use the factor theorem and substitute d(x) into f(x). In this case, substituting x^2 for x in f(x) results in a non-zero polynomial. Therefore, d(x) = x^2 is not a factor of f(x) = 5x^3 + 7x^2 - 7x - 14.
Step-by-step explanation:
To determine if d(x) is a factor of f(x), we can use the factor theorem. The factor theorem states that if x-a is a factor of a polynomial f(x), then f(a) will be equal to zero. In this case, we need to check if d(x) = x^2 is a factor of f(x) = 5x^3 + 7x^2 - 7x - 14. We can substitute x^2 for x in f(x) and check if the result is zero.
Let's substitute x^2 for x in f(x):
f(x) = 5(x^2)^3 + 7(x^2)^2 - 7(x^2) - 14
= 5x^6 + 7x^4 - 7x^2 - 14
Now, we evaluate f(x) at x^2:
f(x^2) = 5(x^2)^6 + 7(x^2)^4 - 7(x^2)^2 - 14
= 5x^12 + 7x^8 - 7x^4 - 14
If d(x) = x^2 is a factor of f(x), then f(x^2) should be equal to zero. However, since f(x^2) = 5x^12 + 7x^8 - 7x^4 - 14 is not equal to zero, d(x) = x^2 is not a factor of f(x) = 5x^3 + 7x^2 - 7x - 14.