Solve the following polynomial inequalities.x3 - 5x2 + 2x ≥ -8

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ishpiro | College Teacher | (Level 1) Educator

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First, rewrite this inequality so that the polynomial function is compared to zero:

`x^3 - 5x^2 + 2x + 8 >=0`

To solve this ineqality, factor the polynomial on the left side. One can notice that -1 is the root of the polynomial because `(-1)^3 - 5*(-1)^2+2*(-1) + 8 = 0` , so x+ 1 is a factor of the polynomial. The rest of the factors can be found by synthetic division:

-1 |   1  - 5  2   8

             -1   6  -8

------------------------

        1   -6   8   0

The first three numbers of the last row are the coefficients of the second factor of the polynomial, `x^2 - 6x + 8` .

The polynomial on the left side is factored as

`(x+1)(x^2 - 6x + 8) = (x+1)(x-2)(x-4)`

To determine whether this expression is greater than 0, note that it will change from positive to negative (of vice versa) at the points -1, 2, and 4.

At the point x = 0, which is on interval (-1, 2), the expression is positive: (0+1)(0 - 2)(0 - 4) = 8

So the polynomial is

negative when x < - 1

positive on (-1, 2)

negative on (2, 4)

positive when x >4

Therefore the solution set of the given inequality is

(-1, 2), x > 4

or in interval notation

`(-1, 2)uu (4, oo).`

` `

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