One factor of f(x) = x² + 9x + 14 is (x + 7). What are all the roots of the function? Use the factor and remainder theorem.

The factor theorem is used to check if a number is the factor of a particular function or not. Let's solve a problem related to this concept.

Answer: The roots of the function f(x) = x² + 9x + 14 are -7 and -2.

Let's understand the solution.

Explanation:

We know that x + 7 is the factor of f(x). Now, by factor theorem, we know x + 7 = 0 is a factor of f(x).

Hence, x = -7 is the root of the given equation.

Now, we divide f(x) = x² + 9x + 14 by (x + 7). We find that (x² + 9x + 14) / (x + 7) = (x + 2) after division. There is no remainder. You can use the long division method to check the answer.

Hence, (x + 2) is also a factor of the equation, and hence, by factor theorem, x + 2 = 0, and x = -2 is the root of f(x).

Thus, the roots of the function f(x) = x² + 9x + 14 are -7 and -2.