Question

Create a 4th degree polynomial that has only the following roots: x=−7,x=3,x=−1/6.(Leave your answer in factored form.)p(x)=Is your polynomial the only 4th degree polynomial that has only these roots? If yes, explain. If no, provide another such polynomial.

Answer 1

`\begin{gathered} p(x)=(x+7)(x+7)(x-3)(6x+1) \\ p(x)=(x+7)(x-3)(x-3)(6x+1) \end{gathered}`

Explanation:

If a 4th degree has only the following roots: x=−7,x=3,x=−1/6.

A 4th-degree polynomial must have 4 roots. What this means is that one of the roots is a repeated root.

If -7 is the repeated root, an example of such polynomial is:

`\begin{gathered} x=-7(\text{twice),x}=3,x=-(1)/(6) \\ x+7=0,x-3=0,x+(1)/(6)=0\implies6x+1=0 \\ p(x)=(x+7)(x+7)(x-3)(6x+1)| \\ \implies p(x)=(x+7)(x+7)(x-3)(6x+1) \end{gathered}`

The given polynomial is not the only 4th-degree polynomial that has only these roots. Any of the given roots can be the repeated root.

If on the other hand, 3 is the repeated root, then p(x) will be:

`p(x)=(x+7)(x-3)(x-3)(6x+1)`