Answer:
[tex]x=-3,\:x=-1,\: x=3,\: x=\frac{9}{2}[/tex]
Step-by-step explanation:
Check rational zeroes with synthetic division
3 | 2 -7 -27 63 81
____ 6 -3 -90 -81__
2 -1 -30 -27 | 0
-3 | 2 -1 -30 -27
____-6_21_27___
2 -7 -9 | 0
-1 | 2 -7 -9
____-2_9
2 -9 | 0
Thus, [tex]2x^4-7x^3-27x^2+63x+81=(x-3)(x+3)(x+1)(2x-9)[/tex]. We have one more zero left, which is solved through the equation [tex]2x-9=0[/tex] by the Zero Product Property:
[tex]2x-9=0\\\\2x=9\\\\x=\frac{9}{2}[/tex]
Therefore, all the zeroes of f(x) from smallest to largest are [tex]x=-3,\:x=-1,\: x=3,\: x=\frac{9}{2}[/tex]
