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
Jun 29, 2018

Answer:

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.