Given a polynomial f(x), if (x + 7) is a factor, what else must be true?

The polynomials are the functions that consist of one or more variables. There are various types of polynomials like quadratic, cubic, etc.

Answer: If (x + 7) is a factor of f(x), then f(-7) = 0 must also be true.

Let's understand the solution in detail.

Explanation:

It is given that (x + 7) is the factor of f(x). Hence, it can be concluded that (x + 7) is a root is f(x).

Then, f(x) must of form (x + 7)(x - a)(x - b)…

Hence, by the factor theorem, we can conclude that f(x) must be zero when x = -7

For instance, let f(x) = x² + 5x -14.

For the above function, if we write it in the product form, we get f(x) = (x + 7)(x - 2).

Hence, (x + 7) is one of the factors of the function.

Now, if we substitute x = -7, then we get f(x) equal to zero. This is the example of the factor theorem (you can also notice that f(2) is also equal to zero here).

Hence, If (x + 7) is a factor of f(x), then f(-7) = 0 must also be true.