Give all possible rational zeros for the following polynomial.
-$$P ( x ) = - 2 x ^ { 4 } + 5 x ^ { 3 } + 6 x ^ { 2 } + 18$$
A) $\pm 1 , \pm 2 , \pm 3 , \pm 6 , \pm 9 , \pm 18$
B) $\pm 1 , \pm \frac { 1 } { 2 } , \pm 2 , \pm 3 , \pm \frac { 3 } { 2 } , \pm 6 , \pm 9 , \pm \frac { 9 } { 2 } , \pm 18$
C) $\pm 1 , \pm \frac { 1 } { 2 } , \pm 2 , \pm 3 , \pm 6 , \pm 9 , \pm 18$
D) $\pm 1 , \pm 2 , \pm \frac { 1 } { 2 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 6 } , \pm \frac { 1 } { 9 } , \pm \frac { 1 } { 18 }$

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Quizplus · Accepted Answer

The Rational Root Theorem states that any rational zero, expressed as a fraction $\frac{p}{q}$ in simplest form, of a polynomial equation with integer coefficients is such that $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient. For the polynomial $P(x) = -2x^4 + 5x^3 + 6x^2 + 18$, the constant term is 18 and the leading coefficient is -2. The factors of 18 are $\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$, and the factors of 2 are $\pm 1, \pm 2$. Therefore, the possible rational zeros include all combinations of the factors of the constant term over the factors of the leading coefficient, which are correctly listed in option B.

Give All Possible Rational Zeros for the Following Polynomial P(x)=−2x4+5x3+6x2+18P ( x ) = - 2 x ^ { 4 } + 5 x ^ { 3 } + 6 x ^ { 2 } + 18P(x)=−2x4+5x3+6x2+18

Give All Possible Rational Zeros for the Following Polynomial $P ( x ) = - 2 x ^ { 4 } + 5 x ^ { 3 } + 6 x ^ { 2 } + 18$