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.

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

  f(x) = ((-x -2)x +5)x -6

step-by-step explanation:

y = | -2(-2) |     →     y = | 4 |     →     y = 4       ⇒   (-2, 4)

y = | -2(-1) |     →     y = | 2 |     →       y =   2       ⇒   (-1, 2)    

y = | -2(0) |     →     y = | 0 |     →       y = 0         ⇒   (0, 0)

y = | -2(1) |   c →     y = | -2 |     →     y =   2         ⇒   (1, 2)    

y = | -2(2) |     →     y = | - 4 |     →     y =   4         ⇒   (2, 4)