Algebra, fundamental theorem

Algebra, fundamental theorem

The following statement is called the fundamental theorem of algebra:

Every polynomial

P.( n ) =x^n+a^n - 1x^n - 1+a^n - 2x^(n - 2)+ ... +a1x +a0?( n ? 1 )

has at least one zero.

This zero does not have to be real.

The discussion of the number of solutions to linear equations, quadratic equations - Domyhomework will do your assignments , cubic equations, etc. is based on the fundamental theorem of algebra:

- Linear equations with one unknown have a solution that can be given after equivalent transformations.

- Quadratic equations have two solutions . Different cases are to be considered:
a) both solutions are real and different,
b) both solutions are real and equal; there is therefore a double-counting solution,
c) there are no solutions in the real-number domain; this case occurs when the discriminant is negative.
(That does not mean, however, that there are no solutions. It is possible to extend the range of real numbers to the range of complex numbers, and in this there are then two solutions.)

- An equation of the third degree (cubic equation) always has at least one real solution; the other two solutions can either also be real and different, real and equal, or non-real.

Is a zero x0 of a polynomial is known - homework help algebra , the polynomial without a remainder is given by (x -x0) divisible. This allows a polynomial of the nth degree to be the product of a polynomial of the (n - 1) th degree and the linear factor (x -x0 ) being represented:

P.n( x ) = ( x -x0) ?P.n - 1( x )

Since the fundamental theorem - accounting homework helper - also applies to the polynomial reduced in degree by 1, one obtains the statement that a polynomial of the nth degree has exactly n zeros.

Useful Resources:

Atoms, structure

Interpretation of line diagrams

The introductory guide to learning programming

Niels Henrik Abel

Geronimo Cardano