What are the upper and lower bounds to find the roots of the function?
f(x
This is made easier IF we can find the real roots right off the bat
So
-3x^3 -6x^2 + 3x + 6 = 0 multiply through by -1
3x^3 + 6x^2 - 3x - 6 = 0 factor
3x^2 ( x - 2) - 3 ( x - 2) = 0
(3x^2 - 3) (x - 2) = 0 so
3x^2 - 3 = 0 x - 2 = 0
3x^2 = 3 x = 2
x^2 = 1
x = ± 1
The roots are -1, 1 , 2
So....if we can perform synthetic division and get alternating signs on the bottom row....then we will have found the lower bound
Test -1 Test - 2 Test - 3
-1 [ 3 6 -3 -6 ] -2 [ 3 6 - 3 - 6 ] - 3 [ 3 6 - 3 - 6 ]
3 -6 -9 9 -18
_____________ ______________ _____________
3 9 NO 3 0 NO 3 -3 6 -24
So x = -3 is the lower bound
Similarly.....if we perfrorm synthetic division and get all postives on the bottom row, then we will have found the upper bound
Test 2
2 [ 3 6 - 3 - 6 ]
6 24 42
_____________
3 12 21 36
So....x = 2 is the upper bound