Solution - Approximation

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (((16 • (x5)) -  (22•5x3)) +  8x) +  1  = 0 
 Step  2  :

Equation at the end of step  2  :

  ((24x5 -  (22•5x3)) +  8x) +  1  = 0 

Step  3  :

Checking for a perfect cube :

 3.1    16x⁵-20x³+8x+1  is not a perfect cube

Trying to factor by pulling out :

 3.2      Factoring:  16x⁵-20x³+8x+1 

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

Group 1:  8x+1 
Group 2:  16x⁵-20x³ 

Pull out from each group separately :

Group 1:   (8x+1) • (1)
Group 2:   (4x²-5) • (4x³)

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 :

 3.3    Find roots (zeroes) of :       F(x) = 16x⁵-20x³+8x+1
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  16  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4 ,8 ,16
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  16x5 - 20x3 + 8x + 1  = 0 

Step  4  :

Equations of order 5 or higher :

 4.1     Solve   16x⁵-20x³+8x+1 = 0

Points regarding equations of degree five or higher.

 (1)  There is no general method (Formula) for solving polynomial equations of degree five or higher.

 (2)  By the Fundamental theorem of Algebra, if we allow complex numbers, an equation of degree  n  will have exactly  n  solutions
(This is if we count double solutions as  2 , triple solutions as  3  and so on

) (3)  By the Abel-Ruffini theorem, the solutions can not always be presented in the conventional way using only a finite amount of additions, subtractions, multiplications, divisions or root extractions

 (4)  If  F(x)  is a polynomial of odd degree with real coefficients, then the equation  F(X)=0  has at least one real solution.

 (5)  Using methods such as the  Bisection  Method, real solutions can be approximated to any desired degree of accuracy.

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) = 16x⁵ - 20x³ + 8x + 1

At   x=   -1.00   F(x)  is equal to  -3.00 
At   x=   0.00   F(x)  is equal to  1.00 

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   -3.000000000  0.000000000    1.000000000
-1.000000000   -3.000000000  0.000000000    1.000000000
-0.500000000   -1.000000000  0.000000000    1.000000000
-0.250000000   -0.703125000  0.000000000    1.000000000
-0.250000000   -0.703125000 -0.125000000    0.038574219
-0.187500000   -0.371871948 -0.125000000    0.038574219
-0.156250000   -0.175196171 -0.125000000    0.038574219
-0.140625000   -0.070261613 -0.125000000    0.038574219
-0.132812500   -0.016307153 -0.125000000    0.038574219
-0.132812500   -0.016307153 -0.128906250    0.011020747
-0.130859375   -0.002671786 -0.128906250    0.011020747
-0.130859375   -0.002671786 -0.129882812    0.004167383
-0.130859375   -0.002671786 -0.130371094    0.000746018
-0.130615234   -0.000963330 -0.130371094    0.000746018
-0.130493164   -0.000108768 -0.130371094    0.000746018
-0.130493164   -0.000108768 -0.130432129    0.000318597
-0.130493164   -0.000108768 -0.130462646    0.000104908
-0.130477905   -0.000001932 -0.130462646    0.000104908
-0.130477905   -0.000001932 -0.130470276    0.000051488
-0.130477905   -0.000001932 -0.130474091    0.000024778
-0.130477905   -0.000001932 -0.130475998    0.000011423
-0.130477905   -0.000001932 -0.130476952    0.000004746

     Next Middle will get us close enough to zero:

     F( -0.130477667 ) is  -0.000000262  

     The desired approximation of the solution is:

       x ≓ -0.130477667

     Note, ≓ is the approximation symbol

One solution was found :