$\"\"$

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The polynomial function is $\"\"$ . \ \

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Identify Possible Rational Zeros :

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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.

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$\"\"$ .

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The factors of leading coefficient is $\"\"$ .

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The factors of the constant term $\"\"$ .

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

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

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$\"\"$

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

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$\"\"$

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Therefore, $\"\"$ is the factor of $\"\"$ .

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The depressed polynomial is $\"\"$ .

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

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$\"\"$

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Therefore, $\"\"$ is the factor of $\"\"$ .

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The depressed polynomial is $\"\"$ .

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Polynomial $\"\"$ is irreducible.

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$\"\"$

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$\"\"$

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To find the complex zeros, equate $\"\"$ to zero.

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$\"\"$ .

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Roots of the quadratic equation $\"\"$ are $\"\"$ .

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$\"\"$

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The complex zeros of $\"\"$ are $\"\"$ .

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$\"\"$

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The complex zeros of $\"\"$ are $\"\"$ .

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By the factor theorem, the factor form of $\"\"$ .

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Factor form of $\"\"$ is $\"\"$ .

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$\"\"$

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The complex zeros are $\"\"$ and $\"\"$ .

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Factor form of $\"\"$ is $\"\"$ .