Root of a function

In mathematics, a root of a function f is a number x that turns the value of f to 0:

[math]\displaystyle{ f(x)=0. }[/math]

for example, one of the roots of sin(x) is [math]\displaystyle{ \pi }[/math], the only root of [math]\displaystyle{ x^n }[/math] is 0, for 2x + 5 it's -2.5 and so on.

Because polynomials are also functions, roots are real things.

The fundamental theorem of algebra says every polynomial with complex coefficients has at least one (complex) root. Using the fundamental theorem of algebra and the polynomial remainder theorem together shows that every complex polynomial of degree n has exactly n complex roots, and some of them may be equal.

root of a polynomial

for first degree polynomials ax + b the root is x = [math]\displaystyle{ \frac{-b}{a} }[/math].

for second degree polynomials ,[math]\displaystyle{ ax^2+bx+c = 0 }[/math] the roots are: [math]\displaystyle{ x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} }[/math].

there are also equations for solving 3rd and 4th degrees but Évariste Galois proved that 5th degree polynomials cannot be solved using radicals.

Polynomial remainder theorem states that if a is a root of a polynomial P(x) If and only if P(x) is divisible by x-a.

the Fundamental theorem of algebra states that the Complex plane is an algebraically close Field meaning that any polynomial from degree n has exactly n roots including repeating roots.

if a Polynomial P(x) = [math]\displaystyle{ (x-a)^t Q(x) }[/math], for t > 0, a is a root of P and a root of the first t-1 derivatives of P.

if P is a polynomial such that all of his coefficients are Integers:

1.if a is an Integer root of P, the constant coefficient is divisible by a.

2.if [math]\displaystyle{ \frac{a}{b} }[/math] is an Rational root of P, the constant coefficient is divisible by a and the leading coefficient is divisible by b.

for example : P(x) = [math]\displaystyle{ x^2 + 2x + 1 }[/math], we can see that the root should be a divisor of 1 meaning it is 1 or -1, after we check the 2 options we can see that the root is -1.