$\"\"$

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Identify Possible Rational Zeros :

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It is not practical to test all possible zeros of a polynomial function using only synthetic substitution.

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The Rational Zero Theorem can be used for finding the some possible zeros to test.

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The function is $\"\"$ .

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

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Therefore the possible rational zeros of $\"\"$ are $\"\"$ .

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$\"\"$

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The function is $\"\"$ .

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Perform the synthetic substitution method by testing $\"\"$ and $\"\"$ .

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$\"\"$

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Since $\"\"$ , conclude that $\"\"$ is a zero of $\"\"$ .

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Therefore, $\"\"$ is a rational zero.

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The depressed polynomial is $\"\"$ .

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$\"\"$

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The depressed polynomial $\"\"$ yeilds no rational zeros.

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Therefore $\"\"$ is a factor of $\"\"$ .

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The final quotient can be written as $\"\"$ .

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The quadratic expression $\"\"$ yeilds no rational zeros.

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Factoring the quadratic expression $\"\"$ .

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By using Factor theorem,

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When $\"\"$ then $\"\"$ is a factor of polynomial.

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Zero is $\"\"$ .

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$\"\"$

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The possible rational zeros of $\"\"$ are $\"\"$ .

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Rational Zero is $\"\"$ .