We can rewrite this function using this rule for quadratics:
#color(red)(x)^2 - color(blue)(y)^2 = (color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y))#
#f(x) = -3x^4(color(red)(x)^2 - color(blue)(9))#
#f(x) = -3x^4(color(red)(x)^2 - color(blue)(3)^2)#
#f(x) = -3x^4(color(red)(x) + color(blue)(3))(color(red)(x) - color(blue)(3))#
Now, we can solve each term on the left side of the function for #0# to find each zero of the function:
Solution 1:
#-3x^4 = 0#
#(-3x^4)/color(red)(-3) = 0/color(red)(-3)#
#(color(red)(cancel(color(black)(-3)))x^4)/cancel(color(red)(-3)) = 0#
#x^4 = 0#
#root(4)(x^4) = root(4)(0)#
#x = 0#
Solution 2:
#x + 3 = 0#
#x + 3 - color(red)(3) = 0 - color(red)(3)#
#x + 0 = -3#
#x = -3#
#x = 0#
Solution 3:
#x - 3 = 0#
#x - 3 + color(red)(3) = 0 + color(red)(3)#
#x - 0 = 3#
#x = 3#
The Solutions Are: #x = {-3, 0, 3}#
The graph of the function touch the #x#-axis at #0#:
graph{y + 3x^6 - 27x^4 = 0 [-10, 10, -50, 400]}
