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?

1 Answer
George C.
Jun 29, 2018

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.

Related questions
See all questions in Intermediate Value Theorem
Impact of this question
3559 views around the world
You can reuse this answer
Creative Commons License