Equation and roots finding without multiplying parenthesis Today I'm studying some functions and I have found that equation:
$f(x) = (x+1)(x+2)(x-3)$
I solve it by multiplying each parenthesis in order to have, after some addition, an equation like that:
$f(x) = x^3-7x-6$
Then I solved it with Ruffini's Theorem and my last equation was this:
$f(x) = (x+1)(x^2-x-6)$
I have read in the web that this procedure is completely useless, because with the first simple equation I could find the roots in a much easier way.
My question is why can I find the roots of an equation by seeing the parenthesis as single equations and without multiplying them other?

A product of real numbers can be zero only if at least one of them is zero. Moreover, if any one of them is zero, the product is always zero. So, you have
$$f(x)=(x+1)(x+2)(x-3)=0$$
You have a product of three terms here - $(x+1), (x+2), (x-3)$. The only way you can have $f(x)=0$ is if either of these is zero. So, the roots are given by
$$x+1=0 \Rightarrow x=-1$$
$$x+2=0 \Rightarrow x=-2$$
$$x-3=0 \Rightarrow x=3$$
In this case, it is not possible for more than one term to be zero simultaneously. However, if there was more than one variable, it may have been possible, leading to more solutions.
