Question 52504
how do you find the real zeros and multiplicity of f(x)=-x^2(x^2-4)(x-5)? How do you determine whether the graph crosses or touches the x-axis at each x-intercept?

find where f(x)=0
we note that f(x)=0.......if x=0
if x^2-4=0...that is (x+2)(x-2)=0......that is x=-2...or....2
if....x-5=0.....that is x=5
so we say it has 4 distint zeros..viz..0,2,-2,5( or roots) and one repeat zero or root equal to '0'of multiplicity =2..since x^2=x*x=0...
if y=0 at more than one point and the curve cuts the x axis at those points,we call those values of x as x intercepts.
if y=0,but the curve has x axis as tangent at that point ,then we say it touches the x axis there.
here the x intercepts are 2,-2,5,where the curve cuts the x axis.the curve touches x axis at x=0

{{{ graph( 500, 500, -5, 10, -100, 100, -(x^2)*(x^2-4)*(x-5)) }}}