If f(1) = 0, what are all the roots of the function f(x)=x^3+3x^2-x-3? use the remainder theorem.

There's no if about it, 



has a zero  so  is a factor.   That's the special case of the Remainder Theorem; since  we'll get a remainder of zero when we divide  by 

At this point we can just divide or we can try more little numbers in the function.  It doesn't take too long to discover  too, so   is a factor too by the remainder theorem.  I can find the third zero as well; but let's say that's out of range for most folks.

So far we have 



where  is the zero we haven't guessed yet.  Again we could divide  by  but just looking at the constant term we must have



so



We check 

We usually talk about the zeros of a function and the roots of an equation; here we have a function  whose zeros are