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: Determinate the roots of the quadradic polynomial $ ax ^ 2 + bx + c $, they 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.

An trivial root is an easily spotted polynomial root. Either because it is the simplest roots like 0, 1, -1, 2 or -2, or because the root is deductible by simply looking at the polynomial.

Example: The polynomial $ (x+3)^2 $ has $ -3 $ as trivial/obvious root

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} $