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.

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

or .

Therefore, the possible rational zeros of are

.

The function is .

Replace zero in the missing term .

Perform the synthetic division method by testing and .

Since ,  conclude that is a zero of .

Therefore is a factor of the polynomial.

The depressed ppolynomial is .

The remaining quadratic factor is can be written as .

The quadratic factor is .

The final quotient can be written as .

yeilds no rational zeros.

Factoring the quadratic expression

By using Factor theorem,

When  then   is a factor of polynomial.

Factored form of .

Zero is .

Therefore the only rational zero of is .

The rational zeros of are .