Polynomial Root

The general principle of root calculation is to evaluate the solutions of the equation polynomial = 0 according to the studied variable (where the curve crosses the y=0 zero axis)..

Example: The roots of the quadradic polynomial $ ax ^ 2 + bx + c $ are the solutions of the equation $ ax ^ 2 + bx + c = 0 $ so $$ x = \frac{ \pm \sqrt{ b ^ 2-4ac } -b}{2 a} $$

The calculation of polynomial roots generally involves the calculation of its discriminant.

Example: For a quadratic polynomial $ ax ^ 2 + bx + c $ the discriminant is $ \Delta = b ^ 2-4 a c $

Use the polynomial discriminant calculator on dCode which automatically adapts to polynomials of degree 2, degree 3, etc. degree n.

A zero of a polynomial function $ P $ is a solution $ x $ such that $ P(x) = 0 $, so it is the other name of a root.

The order of a polynomial (2nd order 2 or quadratic, 3rd order or cubic, 4th order, etc.) is the value of its largest exponent.

Example: $ x^3+x^2+x $ is a polynomial of 3rd order

A polynomial having $ n $ roots / zeros noted $ x_1, x_2, \ cdots, x_n $ is a polynomial of degree $ n $ which can be written in the form: $$ P (x) = (x-x_1)(x-x_2) ... (x-x_n) $$

Example: Find a polynomial having the following roots: $ 1 $ and $ -2 $, answer is written $ P(x) = (x-1)(x+2) $

Sometimes the roots are the same, or the degree is known but there is only one root, then this one is repeated.

Example: Find a polynomial of degree 2 having for unique root $ 1 $, answer is $ P(x) = (x-1)(x-1) = (x-1)^2 = x^2 − 2x + 1 $

The sum of the real roots of a polynomial of degree 2 is $ -\frac{b}{a} $

The product of the real roots of a polynomial of degree 2 is $ \frac{c}{a} $