roots counted with multiplicity, then also, we see that its degree must be greater than or equal to n . In other words, “a polynomial equation of degree n cannot have more than n roots, even if the roots are counted with their multiplicities”. One of the important theorems in the theory of equations is the fundamental theorem of algebra. As the proof is beyond the scope of the Course, we state it without proof.
Theorem . (The Fundamental Theorem of Algebra) Every polynomial equation of degree n ≥ has at least one root in . Using this, we can prove that a polynomial equation of degree n has at least n roots in when the roots are counted with their multiplicities. This statement together with our discussion above says that a polynomial equation of degree n has exactly n roots in when the roots are counted with their multiplicities.
Some authors state this statement as the fundamental theorem of algebra. . . (b) Vieta’s Formula (i) Vieta’s Formula for Polynomial equation of degree Now we obtain these types of relations to higher degree polynomials.
Let us consider a general cubic equation ax bx cx = . By the fundamental theorem of algebra, it has three roots. Let α , β , and γ be the roots. Thus we have - - Theory of Equations ax bx cx = a x )( )( β γ Expanding the right hand side, ax β γ αβ βγ γα αβγ .
Comparing the coefficients of like powers, we obtain β γ = − b a , αβ βγ γα = c a and αβγ = − d a . Since the degree of the polynomial equation is , we have a ¹ and hence division by a is meaningful. If a monic cubic polynomial has roots α , β , and γ , then coefficient of x = − β γ , coefficient of x = αβ βγ γα , and constant term = −αβγ . (ii) Vieta’s Formula for Polynomial equation of