The function is .

Identify Possible Rational Zeros :

Usually 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.

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

Therefore, the possible rational zeros of are .

Consider .

Perform the synthetic substitution method by testing .

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 .

Since , conclude that is a zero of .

Therefore, is a rational zero.

Thus, and are factors of .

The final quotient can be written as .

Factoring the quadratic expression .

By using Factor theorem,

When  then is a factor of polynomial.

Factoring of .

Zeros are .

Therefore the possible rational zeros of are .

The rational zeros of are .