Question

What is discriminant in polynomials ??????

Answer 1

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Answer 2

The Discriminant of A Polynomial is defined as the squares of the product of the distances between roots , when the leading coefficient is unity (one).

$\sf f(x) = a_n{x}^{n} + a_{n - 1} {x}^{n - 1} + ... + a_0$

$a_n \not = 0$

$\sf \\ \therefore \triangle≝ {a}^{2n - 2} \prod _{ i < j \leq n}( x_j - x_i)^{2}$

where, Δ = Discriminant[f(x),x]

The Discriminants for the below polynomial functions (degree. Polynomial name) are:

Special Case

undefined. Zero Polynomial

$f(x) = 0 \\ \triangle ≝ 0$

0. Constant Polynomial

$f(x) = a$

$\implies \triangle = \frac{1}{ {a}^{2} }$

1. Linear Polynomial

$f(x) = ax + b$

$\triangle ≝ 1$

2. Quadratic Polynomial

$f(x) = a {x}^{2} + bx + c$

$\triangle = {a}^{2} (x_2 - x_1) ^{2} \\ = {a}^{2} ((x_1 +x _2) {}^{2} - 4x_1x_2) \\ = {a}^{2} ( \frac{ {b}^{2} }{ {a}^{2} } - 4\frac{ {c} }{a} )$

$\implies \triangle = {b}^{2} - 4ac$

3. Cubic Polynomial

$f(x) = a {x}^{3} + b {x}^{2} + cx + d$

$\triangle = {a}^{4} ((x_3 - x_1)(x_3-x_2) (x_2-x_1))^{2}$

$\triangle= - 27 {a}^{2} {d}^{2} + 18abcd - 4a {c}^{3} - 4 {b}^{3} d + {b}^{2} {c}^{2}$

And, so on...