Question

If a polynomial is of degree n how many real zeros can it have?

Answer 1

Hint: In the given question, we have been asked the number of real zeros a polynomial of degree ‘n’ has. One of the fundamental theorems of algebra states that the polynomial of degree ‘n’ has exactly ‘n’ roots, provided that you can count it by factoring or solving the given polynomial. Thus, a polynomial that does not contain non-constant and have real coefficient can have up to ‘n’ real zeroes.

Complete step-by-step solution:
A polynomial of degree ‘n’ can have at most ‘n’ zeros. As we when dividing the polynomial of ‘n’ degree by the monomial (x – r) where ‘r’ is one of the root of the given polynomial and after dividing you will get \[{{\left( n-1 \right)}^{th}}\]polynomial. And you will continue the process until you will get n = 1.
Here, the question is asking for the real roots.’

Every real root of a polynomial that is having the real coefficient is the root of the factor of that polynomial of degree 1. To find the exact roots is to factor the given polynomial as a product of a simple polynomial of degree 1.

But there are many polynomials of degree 2 that are having only the complex roots i.e. no real roots. Those polynomials will generate non-real roots whose graph cannot touch the x-axis. Therefore,The polynomial that contains or that has real coefficients and that contains non-constant, can have ‘n’ number of roots i.e. exactly the same number of roots as the degree of the polynomial. And,If the polynomial of degree ‘n’ where n is odd then we can say that it will have at least one real root or one real zero.

Note: Students while calculating real roots of any polynomial need to remember that for any non-real complex zeros that will occur in complex conjugate form (i.e. those are in form of iota, an imaginary number or those having root in the denominator) then the possible numbers of real zeros or real roots of that given polynomial is the counting multiplicity is an even number less than ‘n’.