Question

Is x- 10a factor of the function f(x) =x³-75x-250?Use the remainder theorem to justify your answer.

a. Yes, because the remainder when f(x) is divided by x-10is 0.
b. No, because the remainder when f(x) is divided by x-10 is not 0.
c. Insufficient information to determine.
d. Yes, because x-10 is a factor of x³ -75x-250.

Answer 1

No, because the remainder when f(x) is divided by x - 10 is not 0.

Step-by-step explanation:

The remainder theorem states that if a polynomial f(x) is divided by a linear factor x - a, then the remainder will be equal to f(a). In this case, we need to determine if x - 10a is a factor of f(x) = x³ - 75x - 250. To do this, we can use the remainder theorem.

Substitute x = 10a into the function f(x):

f(10a) = (10a)³ - 75(10a) - 250

Simplify the expression:

f(10a) = 1000a³ - 750a - 250

Since the remainder when f(x) is divided by x - 10a is f(10a), we can check if f(10a) equals zero to determine if x - 10a is a factor. If f(10a) = 0, then x - 10a is a factor. If f(10a) ≠ 0, then x - 10a is not a factor.

Therefore, the correct answer is:

b. No, because the remainder when f(x) is divided by x - 10 is not 0.