Polynomials

The canonical form of a polynomial with complex (respectively real) coefficients is P=a_nXⁿ + a_n-1X^n-1 + … + a₀, with a₀, a₁,…, (respectively a₀, a₁, …, a_n) and a_n ≠0. In short we write P = .

If f = a_nXⁿ + … + a₀, with a_n≠0, we say the polynomial f has degree n.

1) Let f $\in$ and z $\in$ . Then f( $z$ ) =

2) Let f $\in$ , a,b such that , b>0

Then f(a±) has the form A±B, A, B

Operations with polynomials

Let f,g , f = and g = , m<n.

The sum of the polynomials f and g is the polynomial written f +g, defined by:

f+g = g+f = , where c_k = equation .

The product of the polynomials f and g is the polynomial , where , k=.

Let f and g be polynomials. Then deg(fg)=degf + degg.

For any polynomials f,g , g≠0, there exist unique polynomials

c, r with the properties:

(1) $f=g\cdot c+r$ ;

(2) deg r < deg g.

*Let f,g . The polynomial f is divisible by the polynomial g if there is a polynomial h such that f=gh. We write f $\vdots$ g or g.

The remainder theorem. The remainder on dividing a polynomial f by the binomial X-a is equal to the value f(a) of the polynomial at a.

Roots of polynomials

Bézout's theorem. Let f be a non-zero polynomial and a $\in$ . Then a is a root of the polynomial f if and only if X-a divides f.

Let f, deg f=2, f=aX²+bX+c. Then:

f is reducible over
f is reducible over $\mathbb{R}$ if and only if $\Delta$ =b²-4ac≥0.

The fundamental theorem of algebra (the d'Alembert–Gauss theorem). Every polynomial equation of degree greater than or equal to 1 has at least one complex root.

Let f . If degf=n, then f has exactly n complex roots (not necessarily distinct).

Let f = , n≥1, with roots x₁, x₂,...,x_n. We have

and the factorisation of f into linear factors is unique.

Viète's relations. Let with roots x₁, x₂ ,...,x_n; we have:

;

Let $\in$ , n $\in$ $\mathbb{N}$ , n≥2 and

Then are the solutions of the equation

We say that $\alpha \in$ is a root of multiplicity p of the polynomial f if divides f and does not divide f.

Let f be a non-zero polynomial with real coefficients and α a root of f.

1) $\alpha$ is also a root of f.

2) α and $\alpha$ have the same multiplicity.

Every polynomial f = of degree n≥1 with real coefficients can be decomposed into a product of first- or second-degree polynomials with real coefficients.

Let f be a non-zero polynomial with rational coefficients and a+b

(a,b,d , d>0, ) a root of f. Then:

1) a-b is also a root of f.

2) a+b and a-b have the same multiplicity.

Let f = be a polynomial of degree n≥1 with integer coefficients, and α= a rational root of f, with p and q $\in \mathbb{Z}$ , q≠0, (p,q)=1. Then:

p divides the constant term a₀
q divides the leading coefficient a_n.