Is x -10 a factor of the function f(x) = x³ + 75x + 250? Explain.

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 – a is a factor of a polynomial, then substituting a into the polynomial should yield zero.

In this case, we need to substitute -10 into the function:

f(-10) = (-10)³ + 75(-10) + 250

Calculating each term:

• (-10)³ = -1000
• 75(-10) = -750
• 250 = 250

Adding these results together:

f(-10) = -1000 - 750 + 250

This simplifies to:

f(-10) = -1500

Since f(-10) = -1500, which is not equal to zero, it follows that x + 10 is not a factor of the function f(x) = x³ + 75x + 250.

In summary, x + 10 is not a factor of the function because substituting -10 into the polynomial does not yield zero.