. . Different types of Polynomial Equations We already know that, for any non–negative integer n , a polynomial of degree n in one variable x is an expression given by P ≡ P(x)= a x a x ... ( ) where a r ∈ are constants, r = , , , with a n ¹ .
The variable x is real or complex. When all the coefficients of a polynomial P are real, we say “ P is a polynomial over ” . Similarly we use terminologies like “ P is a polynomial over ” , “ P is a polynomial over ” , and P is a polynomial over ” . The function P defined by P x a x a x ( ) = is called a polynomial function.
The equation a x a x ... ( ) is called a polynomial equation . If a c a c for some c ∈ , then c is called a zero of the polynomial ( ) and root or solution of the polynomial equation ( ). If c is a root of an equation in one variable x , we write it as“ x is a root”.
The constants a r are called coefficients. The coefficient a n is called the leading coefficient and the term a x n is called the leading term . The coefficients may be any number, real or complex. The only restriction we made is that the leading coefficient a n is nonzero.
A polynomial with the leading coefficient is called a monic polynomial . Remark: We note the following: • Polynomial functions are defined for all values of x . • Every nonzero constant is a polynomial of degree . • The constant is also a polynomial called the zero polynomial ; its degree is not defined.
• The degree of a polynomial is a nonnegative integer. • The zero polynomial is the only polynomial with leading coefficient0 . • Polynomials of degree two are called quadratic polynomials . •