The polynomial function f(x)=−x^3−2x^2+5x−6 has one negative zero.

between what integers does this negative zero occur?

−6 and −5

−5 and −4

−4 and −3

−3 and −2

  -4 and -3

Step-by-step explanation:

You can evaluate the function at the middle of the range of interest, which is [-6, -2], and see which side of that the root lies on.

  f(-4) = ((-(-4) -2)(-4) +5)(-4) -6 = (2(-4) +5)(-4) -6 = (-3)(-4) -6 = 6

Since f(0) = -6, the root lies between -4 and 0. The x-value of -3 further divides the interval [-4, -2] in half, so we can try ...

  f(-3) = ((-(-3) -2)(-3) +5)(-3) -6 = (1(-3) +5)(-3) -6 = 2(-3) -6 = -12

The interval [-4, -3] brackets the root.

_____

We have written the polynomial in "Horner form" to make evaluation easier.

  f(x) = ((-x -2)x +5)x -6

You said                                  -x + 8  >  6

Subtract 8 from each side:      -x        > -2

Add 'x' to each side:                 0        > -2 + x

Add  2  to each side:                      2  >  x

Now, test the choices:

     A).  Can 'x' be 4 ?
            No.  2  is not greater than  4 .

     B).  Can 'x' be  1 ?
           Yes.  2  is greater than  1 .

     C).  Can 'x' be  2 ?
            No.  2  is not greater than  2 .

     D).  Can 'x' be  15 ?
            No.  2  is not greater than  15 .