Question

Find all the zeros of f(x).

Answer 1

Answer:

$x=-3,\:x=-1,\: x=3,\: x=\frac{9}{2}$

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, $2x^4-7x^3-27x^2+63x+81=(x-3)(x+3)(x+1)(2x-9)$. We have one more zero left, which is solved through the equation $2x-9=0$ by the Zero Product Property:

$2x-9=0\\\\2x=9\\\\x=\frac{9}{2}$

Therefore, all the zeroes of f(x) from smallest to largest are $x=-3,\:x=-1,\: x=3,\: x=\frac{9}{2}$