Solution - Approximation

Step by step solution :

Step  1  :

Polynomial Roots Calculator :

 1.1    Find roots (zeroes) of :       F(x) = x³-x-1000
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  -1000.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,5 ,8 ,10 ,20 ,25 ,40 ,50 , etc

 Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

  x3 - x - 1000  = 0 

Step  2  :

Cubic Equations :

 2.1     Solve   x³-x-1000 = 0

Future releases of Tiger-Algebra will solve equations of the third degree directly.

Meanwhile we will use the Bisection method to approximate one real solution.

Approximating a root using the Bisection Method :

We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).

The function is   F(x) = x³ - x - 1000

At   x=   10.00   F(x)  is equal to  -10.00 
At   x=   11.00   F(x)  is equal to  320.00 

Intuitively we feel, and justly so, that since  F(x)  is negative on one side of the interval, and positive on the other side then, somewhere inside this interval,  F(x)  is zero

Procedure :
(1) Find a point "Left" where F(Left) < 0

(2) Find a point 'Right' where F(Right) > 0

(3) Compute 'Middle' the middle point of the interval [Left,Right]

(4) Calculate Value = F(Middle)

(5) If Value is close enough to zero goto Step (7)

Else :
If Value < 0 then : Left <- Middle
If Value > 0 then : Right <- Middle

(6) Loop back to Step (3)

(7) Done!! The approximation found is Middle

Follow Middle movements to understand how it works :

    Left       Value(Left)     Right       Value(Right)

10.000000000  -10.000000000 11.000000000  320.000000000
 0.000000000 -1000.000000000 11.000000000  320.000000000
 5.500000000 -839.125000000 11.000000000  320.000000000
 8.250000000 -446.734375000 11.000000000  320.000000000
 9.625000000 -117.958984375 11.000000000  320.000000000
 9.625000000 -117.958984375 10.312500000   86.397705078
 9.968750000  -19.314483643 10.312500000   86.397705078
 9.968750000  -19.314483643 10.140625000   32.642917633
 9.968750000  -19.314483643 10.054687500    6.441447735
10.011718750   -6.491972268 10.054687500    6.441447735
10.033203125   -0.039155595 10.054687500    6.441447735
10.033203125   -0.039155595 10.043945312    3.197669019
10.033203125   -0.039155595 10.038574219    1.578387914
10.033203125   -0.039155595 10.035888672    0.769399018
10.033203125   -0.039155595 10.034545898    0.365067434
10.033203125   -0.039155595 10.033874512    0.162942351
10.033203125   -0.039155595 10.033538818    0.061889986
10.033203125   -0.039155595 10.033370972    0.011366347
10.033287048   -0.013894836 10.033370972    0.011366347
10.033329010   -0.001264297 10.033370972    0.011366347
10.033329010   -0.001264297 10.033349991    0.005051012
10.033329010   -0.001264297 10.033339500    0.001893354
10.033329010   -0.001264297 10.033334255    0.000314528
10.033331633   -0.000474885 10.033334255    0.000314528
10.033332944   -0.000080179 10.033334255    0.000314528
10.033332944   -0.000080179 10.033333600    0.000117174
10.033332944   -0.000080179 10.033333272    0.000018498
10.033333108   -0.000030840 10.033333272    0.000018498
10.033333190   -0.000006171 10.033333272    0.000018498

     Next Middle will get us close enough to zero:

     F( 10.033333210 ) is  -0.000000004  

     The desired approximation of the solution is:

       x ≓ 10.033333210

     Note, ≓ is the approximation symbol

One solution was found :