Given:
The given function is [tex]f(x)=3x^4-24x[/tex]
We need to determine the zeros of the function.
Zeros of the function:
The zeros of the function are the values that makes the function's value equal to zero.
The zero of the function can be determined by substituting f(x) = 0 in the function.
Thus, we have;
[tex]0=3x^4-24x[/tex]
Switch sides, we get;
[tex]3x^4-24x=0[/tex]
Let us factor out the common term 3x.
Thus, we have;
[tex]3x(x^3-8)=0[/tex]
[tex]3x(x^3-2^3)=0[/tex]
Using the identity, [tex]x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)[/tex], we get;
[tex]3 x(x-2)\left(x^{2}+2 x+4\right)=0[/tex]
Let us solve using the zero factor principle.
Thus, we have;
If [tex]3x=0[/tex] then [tex]x=0[/tex]
If [tex]x-2=0[/tex] then [tex]x=2[/tex]
If [tex]x^2+2x+4=0[/tex] then [tex]x=-1+\sqrt{3} i, x=-1-\sqrt{3} i[/tex] (solving using the quadratic formula)
Thus, the zeros of the function are [tex]x=0, x=2, x=-1+\sqrt{3} i, x=-1-\sqrt{3} i[/tex]
