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Question 499169: Find the greatest lower bound and the smallest upper bound of f(x)=x^4-x^3-12x^2-2x+3
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Find the greatest lower bound and the smallest upper bound of
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f(x)=x^4-x^3-12x^2-2x+3
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....B...
....0..|......1....-1......-12........-2..........3......
....1..|......1......0......-12........-14.....-11.....
....2..|..... 1......1......-10........-22.....-41....
....3..|......1......2.......-6.........-20.....-57....
....4..|......1......3.........0.........-2.......-5.....
....5..|......1......4.........8..........38.......187...(x=5 is smallest upper bound)
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....0..|......1....-1......-12........-2..........3......
..-1..|......1....-2......-10..........8.........-5.....
..-2..|......1....-3.......-6..........10.......-17....
..-3..|......1....-4.........0.........-2......... 9......(x=-3 is the greatest lower bound)
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Upper and Lower Bound Theorem for Real Number Roots.
Using synthetic division to divide f(x) by x-B
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If B>0 and all numbers in the last row from the synthetic division by x-B are positive or zero,
then, B is the smallest upper bound.
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If B<0 and all numbers in the last row from the synthetic division by x-B alternate in sign
then, B is the greatest lower bound. Zero can be considered positive or negative.
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