Fundamental theorem of algebra. In Mathematics and more specifically Higher Algebra , Mathematical Analysis , Geometry, and Complex Variable Functions, it is a theorem that states that every non-constant polynomial of a variable has at least one root . It follows that every polynomial p (x) of a non-constant variable has the same number of real or complex roots as its degree n , a theoretical result that is vital for numerical calculation .
[ hide ]
- 1 Definitions
- 2 Importance
- 3 See also
- 4 References
- 5 Sources
Let be the polynomial of degree n ( n> 0 ) of a variable:
- p (x) = a 0+ a 1 x + a 2 x 2 + … + a n x n .
There is a number r such that p (r) = 0 or what is the same, but expressed as a factorization:
- p (x) = (xr) (b 0+ b 1 x + … + b n-1 x n-1 )
From the last definition it follows that if p (x) can be expressed as:
- p (x) = (xr 1) (b 0 + b 1 x + … + b n-1 x n-1 )
resulting in a new polynomial p 1 (x) :
- p 1(x) = b 0 + b 1 x + … + b n-1 x n-1
grade n-1 ; then the same theorem can be applied to this new polynomial obtaining a new root r 2 so that p (x) could be expressed in the form:
- p (x) = (xr 1) (xr 2 ) p 2 (x)
and thus successively decomposing the resulting lesser polynomials p i (x) ( 0 <i <n ), until we have n roots for p (x) who could be expressed from the product of n binomials of the first degree (xr i ) in the set C of the complexes, as follows:
- p (x) = (xr 1) (xr 2 ) … (xr n )
Although the roots r i ( 0 <i <n + 1 ) have already been mentioned, they can be real or complex and it may be the case that some roots are the same, such as the roots of quadratic polynomials of the perfect square trinomial form a 2 x 2 + 2abx + b 2 or a 2 x 2 -2abx + b 2 .
This direct association between the degree of the polynomial and the number of its roots is of vital importance both for mathematics and for other branches in which the behavior of some phenomenon with polynomials is modeled.
Another aspect is the fact of the parity of complex roots which indicates that:
- If a + bi is a complex root of the polynomial p (x), with rational coefficients, then its conjugate a – bi is also the root of p (x) . This property is not fulfilled when the coefficients are complex a + bi, where a ≠ 0 and b ≠ 0.
This can be seen in the case where the quadratic equations ax 2 + bx + c = 0 , when the discriminant D = b 2 -4ac <0 is calculated , then it is only satisfied with the complexes: