Advanced High-School Mathematics

SECTION 3.7 Discriminant 181

where the convention is that the factor under the” is omitted. From
the above, we see immediately that

f′(αi) =an

∏

j 6 =i

(αi−αj),

and so

R(f,f′) =ann−^1

∏n

i=1

f′(αi) =a^2 nn−^1

∏n

i=1

∏

j 6 =i

(αi−αj) =a^2 nn−^1

∏

j 6 =i

(αi−αj).

Finally, one checks that
∏
j 6 =i

(αi−αj) = (−1)n(n−1)/^2

∏

1 ≤i<j≤n

(αj−αi)^2 ,

which gives the following relationship between the result off(x) and
f′(x) and the discriminant:

Theorem 5.Given the polynomialf(x)with real coefficients, one has
that

R(f,f′) = (−1)n(n−1)/^2 an∆(f).

If we return to the case of the quadraticf(x) =ax^2 +bx+c,then

R(f,f′) = R(ax^2 +bx+c, 2 ax+b)

= det







a b c

2 a b 0

0 2a b







= −(ab^2 − 4 a^2 c) =−a(b^2 − 4 ac).

Since for n = 2 we have (−1)n(n−1)/^2 = −1, we see that, indeed,
R(f,f′) = (−1)n(n−1)/^2 a∆(f) in this familiar case.

Note, finally, that as a result of the representation of ∆(f) in terms
of R(f,f′) we see that ∆(f) is a homogeneous polynomial of degree
2 n−2 in the coefficientsa 0 , a 1 , ..., an.