Find all the rational zeros of the function $f ( x ) = 2 x ^ { 4 } - 9 x ^ { 3 } - 41 x ^ { 2 } + 225 x - 225$.
A) $x = - 3,5 , - 5 , - \frac { 2 } { 3 }$
B) $x = 2 , - 15,5$
C) $x = 3,5 , - 5 , \frac { 3 } { 2 }$
D) $x = - \frac { 3 } { 5 } , \frac { 5 } { 2 } , \frac { 3 } { 2 } , - 5$
E) $x = 2 , - 15 , \frac { 5 } { 2 } , \frac { 3 } { 2 }$

Question

Quizplus · Accepted Answer

To find the rational zeros of the given polynomial $f(x) = 2x^4 - 9x^3 - 41x^2 + 225x - 225$, we can use the Rational Root Theorem. This theorem states that any rational root, expressed in its lowest terms as $\frac{p}{q}$, must have $p$ as a factor of the constant term (-225) and $q$ as a factor of the leading coefficient (2). The factors of -225 are $\pm1, \pm3, \pm5, \pm9, \pm15, \pm25, \pm45, \pm75, \pm225$, and the factors of 2 are $\pm1, \pm2$. By testing these possible rational roots, we find that $x = 3, x = 5, x = -5$, and $x = \frac{3}{2}$ are the roots of the polynomial, which corresponds to option C.

Find All the Rational Zeros of the Function f(x)=2x4−9x3−41x2+225x−225f ( x ) = 2 x ^ { 4 } - 9 x ^ { 3 } - 41 x ^ { 2 } + 225 x - 225f(x)=2x4−9x3−41x2+225x−225

Find All the Rational Zeros of the Function $f ( x ) = 2 x ^ { 4 } - 9 x ^ { 3 } - 41 x ^ { 2 } + 225 x - 225$