Question 636829: X^5-x^4+2x^3-5x^2-13x+20; [1.3, 1.9]?
use the intermediate value theorom to show that the polynomial function has a zero in the given interval
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! What we need to do is find the value of the polynomial when x = 1.3 and when it is 1.9. This can be done the straightforward way by replacing the x's with 1.3's and simplifying:
(1.3)^5-(1.3)^4-5(1.3)^2-13(1.3)+10
and then doing the same with 1.9. But this is easier and faster to use Synthetic Division and the Reminder Theorem. (If you don't know what these are then do it the straightforward way.)
Finding the value of the expression when x = 1.3:
1.3 | 1 -1 2 -5 -13 20
1.3 0.39 3.107 -2.4609 -20.09917
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1 0.3 2.39 -1.893 -15.4609 -0.09917
The lower right corner, -0.09917, is the remainder and, according to the Remainder Theroem, is the value of the polynomial when x = 1.3.
Finding the value of the expression when x = 1.9:
1.9 | 1 -1 2 -5 -13 20
1.9 1.71 7.049 3.8931 -17.30311
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1 0.9 3.71 2.049 -9.1069 2.69689
The lower right corner, 2.69689, is the remainder and, according to the Remainder Theroem, is the value of the polynomial when x = 1.9.
Since the polynomial has a negative value when x = 1.3 and it has a positive value when x = 1.9, then it must have a zero value for an x value somewhere between 1.3 and 1.9.
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