Statement :The third degree polynomial with real coefficients has atleast  nonreal zero.

The above statement is false.

Proof :

Let us consider a polynomial function with a third degree coefficient.

The function is .

Identify Possible Rational Zeros:

It is not practical to test all possible zeros of a polynomial function using only synthetic substitution.

The Rational Zero Theorem can be used for finding the some possible zeros to test.

Let us consider a function as .

Because the leading coefficient is , the possible rational zeros are the integer factors of the constant term .

Therefore the possible rational zeros of are .

The function is .

Perform the synthetic substitution method by testing and .

Since ,  conclude that is a zero of .

Therefore, is a rational zero.

The depressed polynomial is .

Consider .

Perform the synthetic substitution method on the depressed polynomial by testing and .

Since ,  conclude that is a zero of .

Therefore, is a rational zero.

The remaining factor is .

Therefore, and are the factors of .

The final quotient can be written as .

Factoring the quadratic expression .

By using Factor theorem,

When  then   is a factor of polynomial.

Factored form of .

Zeros are .

Therefore the possible rational zeros of are .

The rational zeros are .

The statement is false.