Question

I am trying to solve polynomial inequalities I will include a photo

Answer 1

`(-8,-1)`

Step-by-step explanation

We want to solve the given polynomial:

`2\mleft(x-5\mright)^2\mleft(x+1\mright)\mleft(x+8\mright)<0`

First, let us find the critical points of the polynomial:

`\begin{gathered} (x-5)^2(x+1)(x+8)=0 \\ \Rightarrow x=5,x=5,x=-1,x=-8 \end{gathered}`

Now, we have to test the intervals between the critical points to see which of them will solve the inequality.

CASE 1: x < -8 (Use x = -9)

`\begin{gathered} 2\mleft(-9-5\mright)^2\mleft(-9+1\mright)\mleft(-9+8\mright)<0 \\ 2(-14)^2(-8)(-1)<0 \\ 3136<0 \end{gathered}`

As we can see, this does not work in the original inequality.

CASE 2: -8 < x < -1 (Use x = -6)

`\begin{gathered} 2\mleft(-6-5\mright)^2\mleft(-6+1\mright)\mleft(-6+8\mright)<0 \\ 2(-11)^2(-5)(2)<0 \\ -2420<0 \end{gathered}`

As we can see, this works in the original inequality.

CASE 3: -1 < x < 5 (Use x = 4)

`\begin{gathered} 2\mleft(4-5\mright)^2\mleft(4+1\mright)\mleft(4+8\mright)<0 \\ 2(-1)^2(5)(12)<0^{} \\ 120<0 \end{gathered}`

As we can see, this does not work in the original inequality.

CASE 4: x > 5 (Use x = 10)

`\begin{gathered} 2(10-5)^2(10+1)(10+8)<0 \\ 2(5)^2(11)(18)<0 \\ 9900<0 \end{gathered}`

As we can see, this does not work in the original inequality.

Therefore, the solution of the polynomial inequality given is:

[tex]\begin{gathered} -8