The function is .

The zero is .

Synthetic Substitition :

Perfom the synthetic substitution method to verify is a zero of .

The depressed polynomial is .

Since is a zero of , also a  zero of .

Perfom the synthetic substitution method to verify on the obtained depressed polynomial.

Therefore and are the factors of .

Using these zeros and the new depressed polynomial from the last division, \ \

the polynomial can be written as .

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.

The function is .

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

Therefore, the possible rational zeros of are .

Synthetic Division:

The function is .

Perform the synthetic division method by testing and .

Since ,  conclude that is a zero of .

Therefore is a rational zero.

The new depressed polynomial is .

Finding Rational Zeros :

The remaining depressed polynomial is which does not have rational zeros.

To find the rational zeros use the quadratic formula .

Consider  .

Here .

To find the rational zeros use the quadratic formula .

The zeros of the depressed polynomial are .

Therefore and   are the two factors of the polynomial.

Therefore, are the factors of .

By using Factor theorem,

When  then   is a factor of polynomial.

Factoring of .

So has five real zeros.

Zeros are .

The linear factorization of is  .

Zeros are .