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Polynomial Root

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.

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Polynomial Root -

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# Polynomial Root

## Root Calculator

### What is a polynomial root? (Definition)

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

### How to calculate a 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 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 $\Delta = b^2 - 4 a c$

### How to calculate a discriminant?

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

### How to find trivial roots?

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

### What is a zero for a polynomial?

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.

### What is a nt degree polynomial?

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

### How to find a polynomial with given roots/zeros?

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 $P(x) = (x-1)(x-1) = (x-1)^2 = x^2 − 2x + 1$

### What is the sum of the roots of a 2nd degree polynomial?

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

### What is the product of the roots of a 2nd degree polynomial?

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

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