Solution - Approximation

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                4*(x-3)*3*(2*x^5)-(2*(x-7)*57)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (((4•(x-3))•3)•(2•(x5)))-(2•(x-7)•57)  = 0 

Step  2  :

Equation at the end of step  2  :

  (((4•(x-3))•3)•(2•(x5)))-114•(x-7)  = 0 
 Step  3  :

Equation at the end of step  3  :

  (((4•(x-3))•3)•2x5)-114•(x-7)  = 0 

Step  4  :

Equation at the end of step  4  :

  ((4•(x-3)•3)•2x5)-114•(x-7)  = 0 

Step  5  :

Equation at the end of step  5  :

  (12 • (x - 3) • 2x5) -  114 • (x - 7)  = 0 

Step  6  :

Multiplying exponents :

 6.1    2²  multiplied by  2¹   = 2^{(2 + 1)} = 2³

Equation at the end of step  6  :

  (23•3x5) • (x - 3) -  114 • (x - 7)  = 0 

Step  7  :

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   24x⁶ - 72x⁵ - 114x + 798  = 

  6 • (4x⁶ - 12x⁵ - 19x + 133) 

Checking for a perfect cube :

 8.2    4x⁶ - 12x⁵ - 19x + 133  is not a perfect cube

Trying to factor by pulling out :

 8.3      Factoring:  4x⁶ - 12x⁵ - 19x + 133 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -19x + 133 
Group 2:  4x⁶ - 12x⁵ 

Pull out from each group separately :

Group 1:   (x - 7) • (-19)
Group 2:   (x - 3) • (4x⁵)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 8.4    Find roots (zeroes) of :       F(x) = 4x⁶ - 12x⁵ - 19x + 133
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  4  and the Trailing Constant is  133.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,7 ,19 ,133

 Let us test ....

Note - For tidiness, printing of 19 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  8  :

  6 • (4x6 - 12x5 - 19x + 133)  = 0 

Step  9  :

Equations which are never true :

 9.1      Solve :    6   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Equations of order 5 or higher :

 9.2     Solve   4x⁶-12x⁵-19x+133 = 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= 1.00 and x= 2.00

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) = 4x⁶ - 12x⁵ - 19x + 133

At   x=   2.00   F(x)  is equal to  -33.00 
At   x=   1.00   F(x)  is equal to  106.00 

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)

 2.000000000  -33.000000000  1.000000000  106.000000000
 2.000000000  -33.000000000  0.000000000  133.000000000
 2.000000000  -33.000000000  1.000000000  106.000000000
 2.000000000  -33.000000000  1.500000000   58.937500000
 2.000000000  -33.000000000  1.750000000   17.684570312
 1.875000000   -6.909286499  1.750000000   17.684570312
 1.875000000   -6.909286499  1.812500000    5.647958994
 1.843750000   -0.573370542  1.812500000    5.647958994
 1.843750000   -0.573370542  1.828125000    2.552671126
 1.843750000   -0.573370542  1.835937500    0.993367991
 1.843750000   -0.573370542  1.839843750    0.210911886
 1.841796875   -0.181003099  1.839843750    0.210911886
 1.841796875   -0.181003099  1.840820312    0.015011210
 1.841308594   -0.082981773  1.840820312    0.015011210
 1.841064453   -0.033981734  1.840820312    0.015011210
 1.840942383   -0.009484375  1.840820312    0.015011210
 1.840942383   -0.009484375  1.840881348    0.002763639
 1.840911865   -0.003360312  1.840881348    0.002763639
 1.840896606   -0.000298323  1.840881348    0.002763639
 1.840896606   -0.000298323  1.840888977    0.001232662
 1.840896606   -0.000298323  1.840892792    0.000467170
 1.840896606   -0.000298323  1.840894699    0.000084424
 1.840895653   -0.000106949  1.840894699    0.000084424
 1.840895176   -0.000011263  1.840894699    0.000084424
 1.840895176   -0.000011263  1.840894938    0.000036581

     Next Middle will get us close enough to zero:

     F(  1.840895116 ) is   0.000000698  

     The desired approximation of the solution is:

       x ≓ 1.840895116

     Note, ≓ is the approximation symbol

One solution was found :