Inequality of a polynomial?

Question

Inequality of a polynomial?

Why Do we need to find the real zeros of the polynomial in order to "test" out intervals? I really want to under stand why. Why a problem for example: x^2 - x -6<0= (x+2)(x-3)

Answer 1

Well...

To find x in this, you can either use the quadratic formula, or factorise.. I can see u have factorised here.

In order to factorise, it must equal zero. This is because when factorised one of the factors equals 0. I.E in you equation, x is either -2 or +3...

so for example (x+2) could equal zero. so x is -2    (-2+2) =0!

Same goes for the other

It is impossible to factorise with easy answers unless the quadratic =0.. if it doesnt then u can always rearrange it to do so.

Answer 2

Real zeroes of a polynomial tell you where the function crosses the x-axis, hence where the function is positive or negative.  

Since your critical points for x=3 or x=-2 have to be &lt; 0, you have two options x&lt;3 and x&gt;-2 or x&gt;3 and x&lt;-2.  Testing a point in each interval, ( negative infinity, -2]

            [-2, 3]

and       [3, positive infinity)

yields [-2, 3] as your solution