Given a polynomial f(x), if (x − 4) 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 − 4) is a factor of f(x), then f(4) = 0 must also be true.

Let's understand the solution in detail.

Explanation:

It is given that (x - 4) is the factor of f(x). Hence, it can be concluded that (x - 4) is a root is f(x).

Then, f(x) must of of form (x - 4)(x - a)(x - b)…

Hence, by the factor theorem, we can conclude that f(x) must be zero when x = 4.

For instance, let f(x) = x² - 5x + 4.

For the above function, if we write it in the product form, we get f(x) = (x - 1)(x - 4)

Hence, (x - 4) is one of the factors of the function.

Now, if we substitute x = 4, then we get f(x) equal to zero.

Hence, If (x − 4) is a factor of f(x), then f(4) = 0 must also be true.