Final answer:
Using the Remainder Theorem, f(4) is equal to the remainder when f(x) is divided by (x - 4), thus f(4) = 7.
Step-by-step explanation:
When you divide a polynomial f(x) by (x - 4) and receive a remainder of 7, this tells us that f(4) = 7. This is based on the Remainder Theorem, which states that the remainder of the division of a polynomial by a linear divisor (x - c) is equal to the value of the polynomial evaluated at c, so f(c) is the remainder. Therefore, using the Remainder Theorem, if you substitute 4 into the function f(x), that is, evaluate f(4), you will get the remainder, which is 7.
To find f(4), we need to understand what it means to divide f(x) by (x-4) and get a remainder of 7. This means that when we divide f(x) by (x-4), there will be a remainder of 7. In other words, there exists a polynomial q(x) such that:
f(x) = q(x) · (x-4) + 7
To find f(4), we substitute x = 4 into the equation.
f(4) = q(4) · (4-4) + 7
Since (4-4) = 0, we have:
f(4) = q(4) · 0 + 7
f(4) = 7