Question

How do you use the Intermediate Value Theorem and synthetic division to determine whether or not the following polynomial `P(x) = x^3 - 3x^2 + 2x - 5` have a real zero between the numbers 2 and 3?

Answer 1

Use the Intermediate Value Theorem to find that it does have a zero in `[2, 3]`...

Explanation:

Given:

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

we find:

`P(2) = (color(blue)(2))^3-3(color(blue)(2))^2+2(color(blue)(2))-5 = 8-12+4-5 = -5`

`P(3) = (color(blue)(3))^3-3(color(blue)(3))^2+2(color(blue)(3))-5 = 27-27+6-5 = 1`

So:

`P(2) = -5 < 0 < 1 = P(3)`

The intermediate value theorem tells us that if `f(x)` is continuous on `[a, b]` then `f(x)` takes every value between `f(a)` and `f(b)` somewhere in the interval `[a, b]`.

Hence we can deduce that there is some `x in [2, 3]` such that `P(x) = 0`.

We do not need to use synthetic division.