Factoring a polynomial to get its zeros

Question

While studying about sums and products of roots of polynomials, I found this on the web:

We can take a polynomial, such as: $$f(x) = ax^4 + bx^3 +\dots$$ And then factor it like this: $$f(x) = a(x−p)(x−q)(x−r)\dots$$ Then $p, q, r$, etc are the roots (where the polynomial equals zero)

My question is as follows: Why is there an '$a$' in the given statement $$f(x) = \mathbf a(x−p)(x−q)(x−r)\dots$$

If anything is unclear, ask me to edit the question in comments.

Answer 1

If there wouldn't be an $a$, the leading term (which you get by multiplying all the $x$-es together) would be $x^4$, not $ax^4$.