**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 , 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 , conclude that is a zero of .

Therefore, is a rational zero.

Therefore, and are the 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 .

The possible rational zeros are .

The zeros of are .

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