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Polynomials: Sum and Product of Roots
For the quartic equation 8 p4 – 3 p3 + 8 p2 – 3 p + 2 = 0
the sum of the four roots is and
the product of all the roots is

tickcross


Worked Solution:

First divide the equation by 8, the coefficient of the highest power of p:
p4 –
3
---
8
p3 + p2 –
3
---
8
p +
1
---
4
 = 0
This has the same roots as the original polynomial but now we can equate coefficients in the following:
If the roots are A, B, C, D then
(p – A) (p – B) (p – C) (p – D) = p4 – (A + B + C + D) p3 + ... + ABCD
So the sum of the roots is –coefficient of p3 =
3
---
8
and the product of the roots is constant term =
1
---
4

© MEI Produced by Dr Ron Knott, 13 November 2006
tom . button [AT] mei . org . uk 		Test reference: PolySumProdRoots.html?ref=22734