For a quadratic equation,

ax2 + bx + c = 0,

the quantity b2-4ac is known as the discriminant and indicates what kind of roots the quadratic equation has.

b2-4ac > 0       Two real roots.
b2-4ac = 0       Two repeated roots.
b2-4ac < 0       Complex roots.
As has already been mentioned the quadratic polynomial ax2+ bx + c has discriminant b2 - 4ac. Further the quadratic has repeated roots iff the discriminant vanishes.

Can we generalise this familiar fact to higher degree polynomials? The answer is yes.

For example, a cubic polynomial ax3 + bx2 + cx + d has discriminant b2c2 - 4ac3 - 4b3d -27a2d2 + 18abcd.

In general a polynomial f(x)=a0xm + a1xm-1 + ... + am of degree m has a discriminant D(f) and this discriminant vanishes iff the polynomial has repeated roots.

A formula for the discriminant is easy to give in terms of a certain determinant. To explain this we need to introduce the concept of the resultant of two polynomials. So suppose we have a second polynomial g(x)=b0xn + b1xn-1 + ... + bn then the resultant R(f,g) of the polynomials is the following m+n x m+n determinant:


 --- m --- --- n ---
    
|a a ... a 0 0 ... 0|   
|0 a ... a a 0 ... 0|   
|...................|   
|0 0 ..... a a ... a|   
|b b ....b 0 0 ... 0|
|0 b ..... b 0 ... 0|
|...................|
|0 0 ..... 0 b ... b|

(here you have to read a a ... a as a0 a1 ... am and you have to read b b ... b as b0 b1 ... bn

With these preliminaries behind us we can define the discriminant. D(f)=(-1)m(m-1)/2a0-1R(f,df/dx). If you evaluate the determinant you'll obtain the formulae I gave above in the quadratic and cubic cases.

Dis*crim"i*nant (?), n. [L. discriminans, p. pr. of discriminare.] Math.

The eliminant of the n partial differentials of any homogenous function of n variables. See Eliminant.

 

© Webster 1913.

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