Let f(x) = 2x^3 + x^2 - 5x + c. Given that f(1) = 0 find the values of c.

In order to solve this question we simply must substitute x=1 into f(x). But what is f(x)? What does it actually mean to make a subsitution? f(x) has a fancy name attached to it called a function. It is essentially a machine. It takes in a number, x, and prints out something completely new given some kind of rule. In this case, f(x) is a function because it takes in x and pints out the expression 2x^3 + x^2 - 5x + c. Our task then is to work out what c is provided that f(1) = 0, or more simply put, putting 1 into our function, causes the function to print out a 0. We can alternatively say that we are substituting x = 1 into the formula If we carry out this substitution we see that f(1) = 2(1^3) + (1^3) -5(1) + c = 2(1) + (1) -5(1) + c = 2 + 1 - 5 + c = 3 - 5 + c = c - 2, therefore f(1) = c - 2. We also know from the question provided that f(1) = 0 therefore c - 2 = 0. Finally we can then rearrange this equation in order to get an expression adding 2 to both sides and we get our value for c. The answer is c = 2.