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# What are the upper and lower bounds to find the roots of the function?

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What are the upper and lower bounds to find the roots of the function?

f(x

Oct 3, 2019
edited by Guest  Oct 3, 2019

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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

Oct 3, 2019