For a_nxn+a_n-1xn-1+. s+a_0=0, a_n 0, count all n roots with multiplicity, including complex roots. **Quantity** | **Formula** Sum | r_1+. s+r_n=- a_n-1a_n Product | r_1r_2. s r_n=(-1)n a_0a_n Example For 2x3-7x2+5x+3=0, the sum is 72 and the product is - 32, without solving the cubic.

For \(a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0=0\), \(a_n\ne0\), count all \(n\) roots with multiplicity, including complex roots.

For \(2x^3-7x^2+5x+3=0\), the sum is \(\dfrac72\) and the product is \(-\dfrac32\), without solving the cubic.

Polynomials, Cubic Equations and Partial Fractions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Quantity	Formula
Sum	\(r_1+\cdots+r_n=-\dfrac{a_{n-1}}{a_n}\)
Product	\(r_1r_2\cdots r_n=(-1)^n\dfrac{a_0}{a_n}\)

Sum and product of polynomial roots

Sum and product of polynomial roots