Solve the following polynomial inequality:

2(x^3 - 2x^2 + 3) < x(x-1)(x+1)

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We can distribute on both sides, to get:

`2x^3-4x^2+6<x^3-x`

Subtract `x^3` from both sides, and add `x` to both sides to get:

`x^3-4x^2+x+6 < 0`

One way to do this is to factor the left hand side. You might notice that if you plug in `x=-1` the left hand side is 0. Thus `(x+1)` is a factor:

`(x+1)(ax^2+bx+c)=x^3-4x^2+x+6`

`x*ax^2=x^3` , so we must have `a=1`

`(x+1)(x^2+bx+c) = x^3-4x^2+x+6`

`x*bx+1*x^2 = -4x^2` , so we must have `b=-5`

`(x+1)(x^2-5x+c)=x^3-4x^2+x+6`

`x*c+1*-5x = x` , also, `1*c=6` , thus `c=6`

So:

`(x+1)(x^2-5x+6)<0`

But we can factor further:

`x^2-5x+6 = (x-2)(x-3)`

So:

`(x+1)(x-2)(x-3)<0`

`x=-1` , `x=2` , and `x=3` are places where the left hand side is zero, so they are places where there is a switch from negative to positive, or positive to negative.

So we look at: `x<-1` , `-1<x<2` , `2<x<3` , and `x>3`

One at a time:

`x<-1`

Then:

`x+1<0` , `x-2 < -3 < 0` . `x-3<-4<0`

So we have a negative times a negative times a negative, which is negative.

So if `x<-1` then `(x+1)(x-2)(x-3)<0` , so `2(x^3-2x^2+3)<x(x-1)(x+1)`

Next up:

`-1<x<2`

Then:

` ``x+1 > 0` , `x-2 < 0` , `x-3<-1<0`

So we have a positive times a negative times a negative, which is positive. Thus it isn't true that `(x+1)(x-2)(x-3)<0 `

Now we try `2<x<3`

Then `x+1>0` , `x-2>0` , and `x-3 < 0`

So we have a positive times a positive times a negative, which is negative. So it is true that ` `

So if 2<x<3 then `` , so ``

Finally, suppose x>3

Then x+1, x-2, and x-3 are all positive. So multiplying them all together gives you a positive, which doesn't satisfy the inequality.

So the inequality works if x< -1 or if 2<x<3

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