. . Vieta’s formula for Polynomial Equations What we have learnt for quadratic polynomial, can be extended to polynomials of higher degree. In this section we study the relations of the zeros of a polynomial of higher degree with its coefficients.
We also learn how to form polynomials of higher degree when some information about the zeros are known. In this chapter, we use either zeros of a polynomial of degree n or roots of polynomial equation of degree n . . .
(a) The Fundamental Theorem of Algebra If a is a root of a polynomial equation P x ( ) = , then ( is a factor of P x ( ). So, deg( ( )) P x ≥ . If a and b are roots of P x ( ) = then ( )( is a factor of P x ( )and hence deg ( ( )) P x ≥ . Similarly if P x ( ) = has n roots, then 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 . In earlier classes we have learnt about “multiplicity”. Let us recall what we mean by “multiplicity”. We know if ( a k is a factor of a polynomial equation P x ( ) = and ( a k + is not a factor of the polynomial equation, P x ( ) = , then a is called a root of multiplicity k .
For instance, is a root of multiplicity for the equation x and x . Though we are not going to use complex numbers as coefficients, it is worthwhile to mention that the imaginary number + i is a root of multiplicity for the polynomials x i x + + and x = . If a is a root of multiplicity for a polynomial equation, then a is called a simple root of the polynomial equation. If P x ( ) = has n