How do you use synthetic division to find all the rational zeroes of the function #f(x)= 2x^3 - 3x^2-11x+6#?

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Answer 1 · 2015-08-20T03:02:48

#3, -2, 1/2#

A rational zero must be #± frac{\text{a divisor of 6}}{\text{a divisor of 2}} = ± frac{{1,2,3,6}}{{1,2}} \in ± {1,2,3,6, 1/2, 2/2, 3/2, 6/2}#

#f(1) = 2 - 3 - 11 + 6 < 0#

#f(2) = 16 - 12 - 22 + 6 < 0#

#f(3) = 54 - 27 - 33 + 6 = 0#

By Briot-Ruffini,
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#\frac{f(x)}{x - 3} = 2x^2 + 3x - 2 = 0#

#\Delta = 9 + 4 * 2 * 2 = 25#

#x = frac{- 3 ± 5}{4}#