Find the upper and lower bounds of the real zeros

Question

D(x)=2x^4 -9x^3 +2x^2 +9x -4

Use synthetic division to identify the best integer upper and lower bounds of the real zeros. (Although there may be more than one correct bound, you must find the BEST integer value- I.e. The lowest upper bound and the highest lower bound)

Answer 1

5⌋     2       -9      2      9      -4

   _________10__ 5___35__220

        2         1     7     44    216

 

Since all of the numbers in the bottom row are positive, 5 is an upper bound for the real zeros of f(x).

 

-1⌋    2       -9      2       9     -4

    ________-2___11___-13___4

        2       -11    13      -4     0

 

Since the numbers in the bottom row alternate in sign (0 counts as a sign change), -1 is a lower bound for the real zeros of f(x).

 

So, all of the real zeros of f(x) lie in the interval [-1, 5).    

    

Answer 2

Newton's method will be applied to solve the problem 

D'(x) = 8x^3-27X^2+4X+9=0

 

so X1 = X0 -D(x)/D'(x)

 

In this case we use a random number and solve for x1. If X1 - X0 is acceptable error then we accept it other wise we take value of new X1 as X and repeat the process till we get the solution. 

 

 

 

https://en.wikipedia.org/wiki/Newton%27s_method