Vieta's Formulas can be used to relate the sum and product of the roots of a polynomial to its coefficients. The simplest application of this is with quadratics. If we have a quadratic x2 + ax + b = 0 with solutions p and q, then we know that we can factor it as
x2 + ax + b = (x-p)(x-q), Note that the first term is x2, not ax2.
Using the distributive property to expand the right side we get
x2 + ax + b = x2 - (p+q)x + pq
We know that two polynomials are equal if and only if their coefficients are equal, so the above means that
a = -(p+q), and
b = pq.
In other words, the product of the roots is equal to the constant term, and the sum of the roots is the opposite of the coefficient of the x term.