Answer: \(\pm1 , \pm2 , \pm3 , \pm6 , \pm9 , \pm 18, \pm \dfrac{1}{2}, \pm \dfrac{3}{2}, \pm\dfrac{9}{2}\)
Step-by-step explanation:
- The Rational Zeroes theorem says that if f(x) has integer coefficients and is a rational zero, then q is the factor of leading coefficient and p is the factor of constant term.
Given polynomial= \(f(x) = -2x^4 + 4x^3 + 3x^2 + 18\)
leading coefficient : q = -2
constant term p = 18
Now, Factors of q = \(\pm1, \pm2\)
Factors of p= \(\pm1, \pm2, \pm3 ,\pm 6, \pm9, \pm18\)
Then , by rational root theorem the rational roots are in the form :
\(\dfrac{p}{q}=\dfrac{\pm1}{1}, \dfrac{\pm1}{2} , \dfrac{\pm2}{1}, \dfrac{\pm2}{2}, \dfrac{\pm3}{1}, \dfrac{\pm3}{2}, \dfrac{\pm6}{1}, \dfrac{\pm6}{2}, \dfrac{\pm9}{1}, \dfrac{\pm9}{2}, \dfrac{\pm18}{1}, \dfrac{\pm18}{2}\\\\\Rightarrow\ \dfrac{p}{q}=\pm1 , \pm2 , \pm3 , \pm6 , \pm9 , \pm 18, \pm \dfrac{1}{2}, \pm \dfrac{3}{2}, \pm\dfrac{9}{2}\)
Hence , the list of all possible rational zeros of the function are :
\(\pm1 , \pm2 , \pm3 , \pm6 , \pm9 , \pm 18, \pm \dfrac{1}{2}, \pm \dfrac{3}{2}, \pm\dfrac{9}{2}\)
