Find all the rational zeros of the function $f ( x ) = 2 x ^ { 5 } + x ^ { 4 } - 11 x ^ { 3 } + 7 x ^ { 2 } - 13 x + 6$ .
A) $x = - \frac { 1 } { 2 } , - 3 , - 1$
B) $x = \frac { 2 } { 3 } , 1,2$
C) $x = \frac { 1 } { 2 } , - \frac { 3 } { 2 } , 2$
D) $x = \frac { 1 } { 2 } , - \frac { 3 } { 2 }$
E) $x = \frac { 1 } { 2 } , - 3,2$

Question

Quizplus · Accepted Answer

To find the rational zeros of the polynomial $f(x) = 2x^5 + x^4 - 11x^3 + 7x^2 - 13x + 6$, we use the Rational Root Theorem. This theorem states that any rational root, expressed in its lowest terms $\frac{p}{q}$, is such that $p$ is a factor of the constant term (in this case, 6) and $q$ is a factor of the leading coefficient (in this case, 2). The factors of 6 are $\pm1, \pm2, \pm3, \pm6$, and the factors of 2 are $\pm1, \pm2$. This gives us potential rational roots of $\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{6}{2}$. Testing these values or using synthetic division, we find that $x = \frac{1}{2}, -3, 2$ are indeed zeros of the polynomial, matching choice E.

Find All the Rational Zeros of the Function f(x)=2x5+x4−11x3+7x2−13x+6f ( x ) = 2 x ^ { 5 } + x ^ { 4 } - 11 x ^ { 3 } + 7 x ^ { 2 } - 13 x + 6f(x)=2x5+x4−11x3+7x2−13x+6

Find All the Rational Zeros of the Function $f ( x ) = 2 x ^ { 5 } + x ^ { 4 } - 11 x ^ { 3 } + 7 x ^ { 2 } - 13 x + 6$