Why does the test for an upper bound on the zeros using synthetic division not work on a particular function?

Question

y= x⁷ -5x⁵ +2x⁴ +12x² -6x-12

When I graph this the highest zero is 1.72, When I do the upper bound test using synthetic division with the value of 2, the third term comes out negative when all the terms should be positive. I have never seen this test not work and am not sure what is happening.

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Answer 1

When I do synthetic division using 1.72125 as a solution, I get all positive coefficients for the quotient and a positive remainder. If 1.72125 is the upper bound on the zeros then you must use 1.72125 when performing synthetic division (not 2). Also, for a function like this, you should carry at least 4 decimal places on your calculations. Lastly, make sure that your coefficients are in descending order with respect to the degrees of the variable. You should set up your synthetic division like this:

1.72125 | 1 0 -5 2 0 12 -6 -12

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Coefficients of quotient: 1.72125, 2.9627015, 0.09955006, 2.1713505, 3.73743713, 18.43306366, 25.72791

Remainder: 32.2841665