Take the stipulated polynomial roots . Their first three elementary symmetric polynomials are
(1)
Their monic polynomial is therefore
(2)
The constant coefficient is , while the product of the three roots of a polynomial is : for a cubic polynomial, these have opposite signs. Expanding the factors directly verifies the same polynomial identity.
Figure 1.
The roots determine the signed coefficients of a cubic
. An original exact calculation: the graph shows the stipulated polynomial, and the ledger groups choices of one, two or three root positions into elementary symmetric polynomials. The alternating factors convert these values into monic polynomial coefficients. Details.
For the nonmonic polynomial , the same ratios give sum , pairwise-product sum , and product . It is times the monic polynomial above, so its roots of a polynomial are the same. Checking only the sum and product would leave the middle elementary symmetric polynomial unchecked; all three are needed to recover a monic polynomial of polynomial degree three.

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