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.
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.