Relating roots and coefficients of quadratics

When the roots of a polynomial are known, we can multiply out factors of the form (z − r_k) to obtain the coefficients of the polynomial. This leads to certain relationships between the roots and coefficients. Consider the particular case of a quadratic polynomial, that is, one of degree 2. If the original quadratic is az² + bz + c, then the factored form will be given by a(z − r)(z − s) where r and s are the roots. By equating these two different forms for the polynomial, we are led to

.

Now multiplying out the two factors on the left produces

This shows that −(r + s) = b/a and rs = c/a.

A similar approach can be pursued for a polynomial of any degree. In general, for a polynomial of degree n, say a_n zⁿ + a_{n − 1} z^{n − 1} + ··· + a₁ z + a₀ , the sum of the roots is equal to −a_{n − 1}/a_n, and the product of the roots is given by (−1)ⁿa₀ /a_n. There are also formulas for the other coefficients, but we will not need them for this article. For further information, see the article on elementary symmetric polynomials in Wikipedia.