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.

The function is .

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

Therefore, the possible rational zeros of are

The function is .

Perform the synthetic substitution method by testing and .

Since is , 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 is conclude that is a zero of .

Therefore, is a rational zero.

The new depressed polynomial is .

Consider .

Perform the synthetic substitution method on the new depressed polynomial by testing

and .

Since is conclude that is a zero of .

Therefore, is a repeated rational zero.

Therefore and are the factors of .

The final quotient can be written as .

The factor does not have rational zeros.

By using Factor theorem,

When  then   is a factor of polynomial.

Factoring of .

Zeros are .

The possible rational zeros are

.

The zeros of are .