$\"\"$

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

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$\"\"$

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The depressed polynomial is $\"\"$ .

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

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$\"\"$

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Factor the polynomial $\"\"$ .

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$\"\"$

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Therefore, $\"\"$ and $\"\"$ are also the factors of $\"\"$ .

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$\"\"$

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By factor theorem, when $\"\"$ then $\"\"$ is a factor of polynomial.

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So, $\"\"$ .

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$\"\"$ , and $\"\"$ .

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Thus, the real zeros $\"\"$ are $\"\"$ .

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

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$\"\"$

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The real zeros $\"\"$ are $\"\"$ .

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