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 equation is .

The equation can be written as .

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

or

.

Consider .

Perform the synthetic substitution method  by testing  and .

By using synthetic substitution, it can be determined that  is a rational zero.

The depressed polynomial is .

Consider .

Perform the synthetic substitution method of the obtained depressed polynomial by testing

and .

By using synthetic substitution, it can be determined that  is a rational zero.

The new depressed polynomial is .

Consider .

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

and .

By using synthetic substitution, it can be determined that  is a rational zero.

Therefore, are the factors of the equation.

The final quotient can be written as .

.

Thus, the zeros are .

The zeros of the equation are .