Question

Find all zeros of ƒ(x) = –2x3(x + 2)2. Then determine the multiplicity at each zero. State whether the graph will touch or cross the x-axis at the zero.

Answer 1

`\\ewline{\text{f(x) has roots 0 multiplicity 3 and -2 multiplicity 2}}`
`\bigskip\\ewline{\text{The root 0 has an odd multiplicity so it will cross the x-axis at the zero}}`
`\bigskip\\ewline{\text{The root -2 has an even multiplicity so it willtouch the x-axis at the zero}}`

Explanation:

`\\ewline{\text{Assuming you mean }f(x)=-2x^3(x+2)^2 }`

`\bigskip\\ewline{\text{If the multiplicity is even, then the graph will touch the x-axis at the zero}}`
`\bigskip\\ewline{\text{If the mutlpilcty is odd, then the graph will cross the x-axis at that zero}}`
`\bigskip\\ewline{\text{Given } f(x)=-2x^3(x+2)^2}`
`\bigskip\\ewline{\text{it can be factored into } f(x)=-2(x-0)^3(x-(-2))^2}`
`\bigskip\\ewline{\text{Therefor f(x) has roots 0 multiplicity 3 and -2 multiplicity 2}}`
`\bigskip\\ewline{\text{The root 0 has an odd multiplicity so it will cross the x-axis at the zero}}`
`\bigskip\\ewline{\text{The root -2 has an even multiplicity so it willtouch the x-axis at the zero}}`