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Question

Given the function f(x)=x(x+2)^4(x-4)^4(x-2)^3(x+5)^3, list the zeros of f(x), corresponding multiplicities, and whether they touch or cross the x-axis

Answer 1

It looks like your polynomial is in factored form, so...

Set each factor to 0

x = 0 -> x =0 multiplicity of 1 cross x-axis

(x+2) = 0 -> x = -2 multiplicity of 4 touch x-axis

(x-4) = 0 -> x = 4 multiplicity of 4 touch x-axis

(x-2) =0 -> x = 2 multiplicity of 3 cross x-axis

(x+5) = 0 -> x = -5 multiplicity of 3 cross x-axis