Tool to calculate/find the root of a polynomial. In mathematics, a root of a polynomial is a value for which the polynomial is 0. A polynomial of degree n can have between 0 and n roots.

Polynomial Root - dCode

Tag(s) : Functions

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The roots of a polynomial $ P(x) $ whose values of $ x $ for which the polynomial is worth $ 0 $ (ie $ P(x) = 0 $).

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 quadratic polynomial $ ax^2 + bx + c $, they are the solutions of the equation $ ax^2 + bx + c = 0 $ so $$ x=\frac{ \pm \sqrt{b^2-4 a c}-b}{2 a} $$

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

__Example:__ For a quadratic polynomial of the form $ ax^2 + bx + c $ the discriminant formula 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 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 trivially deductible.

__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: $$ P(x) = (x-x_1)(x-x_2) \cdots (x-x_n) $$

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

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*Polynomial Root* on dCode.fr [online website], retrieved on 2023-09-21,

- Root Calculator
- What is a polynomial root? (Definition)
- How to calculate a polynomial root?
- How to calculate a discriminant?
- How to find trivial roots?
- What is a zero for a polynomial?
- What is a nt degree polynomial?
- How to find a polynomial with given roots/zeros?
- What is the sum of the roots of a 2nd degree polynomial?
- What is the product of the roots of a 2nd degree polynomial?

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