Question

For the polynomial function ƒ(x) = x^4 −25x^2, find the zeros. Then determine the multiplicity at each zero and state whether the graph displays the behavior of a touch or a cross at each intercept.x = 0, cross; x = −5, cross; x = 5, touchx = 0, cross; x = −5, cross; x = 5, crossx = 0, touch; x = −5, cross; x = 5, crossx = 0, cross; x = −5, touch; x = 5, touch

Answer 1

We are given

`\begin{gathered} f(x)=x^4-25x^2 \\ \\ f(x)=x^2(x^2-25) \\ \\ f(x)=x^2(x-5)(x+5) \end{gathered}`

The zeros are

`\begin{gathered} x=0\text{ with multiplicity 2} \\ x=5\text{ with multiplicity 1} \\ x=-5\text{ with multiplicity 1} \end{gathered}`
`\begin{gathered} x=0\text{ touch} \\ x=5\text{ cross} \\ x=-5\text{ cross} \end{gathered}`