= − ∑ α α α

= − ∑ α α α coefficient of x = ∑ − n = ∑ α α ... coefficient of x = constant term ∑ denotes the sum of product of all roots taken two at a time, α α α ∑ denotes the sum of product of all roots taken three at a time, and so on. If α β γ , , and δ are the roots of a quartic equation, then ∑ and so on. Thus we have, β γ δ αβ αβ αγ αδ βγ βδ γδ αβγ αβγ αβδ αγδ βγδ αβγδ ∑ = βγδ When the roots are available in explicit numeric form, then also we use these convenient notations.

We have to be careful when handling roots of higher multiplicity. For instance, if the roots of a cubic equation are , , , then ∑ . From the above discussion, we note that for a monic polynomial equation, the sum of the roots is the coefficient of x n − multiplied by − and the product of the roots is the constant term multiplied by − n . Example .

If α , β , and γ are the roots of the equation x px qx , find the value of βγ ∑ in terms of the coefficients. Since α , β , and γ are the roots of the equation x px qx , we have