To determine if x + 10 is a factor of the polynomial function f(x) = x³ + 75x + 250, we can use the Factor Theorem. According to this theorem, if x + 10 is a factor, then the function evaluated at x = -10 should equal zero.
First, let’s substitute -10 into the function:
f(-10) = (-10)³ + 75(-10) + 250
Calculating this step-by-step:
- (-10)³ = -1000
- 75(-10) = -750
- 250 remains the same.
Now, adding these results together:
f(-10) = -1000 – 750 + 250
Now, let’s simplify:
f(-10) = -1000 – 750 = -1750
-1750 + 250 = -1500
Since f(-10) = -1500, which is not equal to zero, we can conclude that x + 10 is not a factor of the function f(x) = x³ + 75x + 250.
In summary, for x + 10 to be a factor, substituting -10 into the function must yield zero, which it does not in this case. Therefore, x + 10 is not a factor of the polynomial.