Solution - Approximation

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((((((2•(x6))-(9•(x5)))-(10•(x4)))-(3•(x3)))+(2•5x2))-9x)+2  = 0 
 Step  2  :

Equation at the end of step  2  :

  ((((((2•(x6))-(9•(x5)))-(10•(x4)))-3x3)+(2•5x2))-9x)+2  = 0 
 Step  3  :

Equation at the end of step  3  :

  ((((((2•(x6))-(9•(x5)))-(2•5x4))-3x3)+(2•5x2))-9x)+2  = 0 
 Step  4  :

Equation at the end of step  4  :

  ((((((2•(x6))-32x5)-(2•5x4))-3x3)+(2•5x2))-9x)+2  = 0 
 Step  5  :

Equation at the end of step  5  :

  (((((2x6-32x5)-(2•5x4))-3x3)+(2•5x2))-9x)+2  = 0 

Step  6  :

Polynomial Roots Calculator :

 6.1    Find roots (zeroes) of :       F(x) = 2x⁶-9x⁵-10x⁴-3x³+10x²-9x+2
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  2  and the Trailing Constant is  2.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,2

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  6  :

  2x6 - 9x5 - 10x4 - 3x3 + 10x2 - 9x + 2  = 0 

Step  7  :

Equations of order 5 or higher :

 7.1     Solve   2x⁶-9x⁵-10x⁴-3x³+10x²-9x+2 = 0

In search of an interavl at which the above polynomial changes sign, from negative to positive or the other wayaround.

Method of search: Calculate polynomial values for all integer points between x=-20 and x=+20

Found change of sign between x= 0.00 and x= 1.00

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) = 2x⁶ - 9x⁵ - 10x⁴ - 3x³ + 10x² - 9x + 2

At   x=   1.00   F(x)  is equal to  -17.00 
At   x=   0.00   F(x)  is equal to  2.00 

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)

 1.000000000  -17.000000000  0.000000000    2.000000000
 1.000000000  -17.000000000  0.000000000    2.000000000
 0.500000000   -1.250000000  0.000000000    2.000000000
 0.500000000   -1.250000000  0.250000000    0.280761719
 0.375000000   -0.385887146  0.250000000    0.280761719
 0.312500000   -0.047817111  0.250000000    0.280761719
 0.312500000   -0.047817111  0.281250000    0.115604820
 0.312500000   -0.047817111  0.296875000    0.033915055
 0.304687500   -0.006914365  0.296875000    0.033915055
 0.304687500   -0.006914365  0.300781250    0.013505517
 0.304687500   -0.006914365  0.302734375    0.003297364
 0.303710938   -0.001807992  0.302734375    0.003297364
 0.303710938   -0.001807992  0.303222656    0.000744806
 0.303466797   -0.000531562  0.303222656    0.000744806
 0.303466797   -0.000531562  0.303344727    0.000106629
 0.303405762   -0.000212465  0.303344727    0.000106629
 0.303375244   -0.000052917  0.303344727    0.000106629
 0.303375244   -0.000052917  0.303359985    0.000026856
 0.303367615   -0.000013031  0.303359985    0.000026856
 0.303367615   -0.000013031  0.303363800    0.000006913
 0.303365707   -0.000003059  0.303363800    0.000006913

     Next Middle will get us close enough to zero:

     F(  0.303365231 ) is  -0.000000566  

     The desired approximation of the solution is:

       x ≓ 0.303365231

     Note, ≓ is the approximation symbol

One solution was found :