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Determine all the possibilities for rational zeros. (Enter your answers as a comma-separated list.)

\[7x^5 - 27x^3 + 27x^2 + 12x - 2 = 0\]

Find a third-degree polynomial \(P(x)\) that has zeros at \(x = -3\), \(x = 3\), and \(x = 4\), and whose \(x\)-term has a coefficient of 45.

Answer :

Final answer:

The possibilities for rational zeros are ±1, ±2, ±1/7, and ±2/7.

Explanation:

To determine the possibilities for rational zeros, we can use the Rational Root Theorem. The theorem states that if a polynomial has a rational root, it can be written as a fraction where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient.

For the given polynomial, the constant term is -2 and the leading coefficient is 7. The factors of -2 are ±1 and ±2, and the factors of 7 are ±1 and ±7. Therefore, the possible rational zeros are: ±1, ±2, ±1/7, ±2/7.

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