To find all the zeros of the polynomial function
[tex]\[ f(x) = 2x^4 - 7x^3 - 27x^2 + 63x + 81, \][/tex]
we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
Let's start by finding the roots of the polynomial:
[tex]\[ 2x^4 - 7x^3 - 27x^2 + 63x + 81 = 0. \][/tex]
By solving this equation, we determine the zeros of [tex]\( f(x) \)[/tex].
The roots of the polynomial equation are:
[tex]\[ x = -3, \][/tex]
[tex]\[ x = -1, \][/tex]
[tex]\[ x = 3, \][/tex]
[tex]\[ x = 4.5. \][/tex]
These are the solutions where the polynomial equals zero.
When we arrange the roots from smallest to largest, we get:
[tex]\[ -3, -1, 3, 4.5. \][/tex]
Therefore, the solutions are:
[tex]\[ x = -3, -1, 3, 4.5. \][/tex]
So, the complete solution with the zeros listed in order from smallest to largest is:
[tex]\[ x = [-3, -1, 3, 4.5]. \][/tex]
