Solution - Approximation

Step by step solution :

Step  1  :

              3000  
 Simplify   ————————
            (1 + x)3

Equation at the end of step  1  :

    3000      
  ———————— -  10000  = 0 
  (x + 1)3    

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  (x+1)³   as the denominator :

             10000     10000 • (x + 1)3
    10000 =  —————  =  ————————————————
               1           (x + 1)3    

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 3000 - (10000 • (x+1)3)     -10000x3 - 30000x2 - 30000x - 7000
 ———————————————————————  =  ——————————————————————————————————
       1 • (x+1)3                 1 • (x3 + 3x2 + 3x + 1)      

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -10000x³ - 30000x² - 30000x - 7000  = 

  -1000 • (10x³ + 30x² + 30x + 7) 

Checking for a perfect cube :

 3.2    10x³ + 30x² + 30x + 7  is not a perfect cube

Trying to factor by pulling out :

 3.3      Factoring:  10x³ + 30x² + 30x + 7 

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

Group 1:  30x + 7 
Group 2:  30x² + 10x³ 

Pull out from each group separately :

Group 1:   (30x + 7) • (1)
Group 2:   (x + 3) • (10x²)

Bad news !! Factoring by pulling out fails :

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

Checking for a perfect cube :

 3.4    Factoring:  x³ + 3x² + 3x + 1 
 .

  x³ + 3x² + 3x + 1  is a perfect cube which means it is the cube of another polynomial 

 In our case, the cubic root of  x³ + 3x² + 3x + 1  is  x + 1  

 Factorization is  (x + 1)³

Polynomial Roots Calculator :

 3.5    Find roots (zeroes) of :       F(x) = 10x³ + 30x² + 30x + 7
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  10  and the Trailing Constant is  7.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,5 ,10
 of the Trailing Constant :  1 ,7

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  -1000 • (10x3 + 30x2 + 30x + 7)
  ———————————————————————————————  = 0 
             (x + 1)3            

Step  4  :

When a fraction equals zero :

 4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  -1000•(10x3+30x2+30x+7)
  ——————————————————————— • (x+1)3 = 0 • (x+1)3
          (x+1)3         

Now, on the left hand side, the  (x+1)³   cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -1000  •  (10x³+30x²+30x+7)  = 0

Equations which are never true :

 4.2      Solve :    -1000   =  0

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

Cubic Equations :

 4.3     Solve   10x³+30x²+30x+7 = 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) = 10x³ + 30x² + 30x + 7

At   x=   -1.00   F(x)  is equal to  -3.00 
At   x=   0.00   F(x)  is equal to  7.00 

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    7.000000000
-1.000000000   -3.000000000  0.000000000    7.000000000
-0.500000000   -1.750000000  0.000000000    7.000000000
-0.500000000   -1.750000000 -0.250000000    1.218750000
-0.375000000   -0.558593750 -0.250000000    1.218750000
-0.375000000   -0.558593750 -0.312500000    0.249511719
-0.343750000   -0.173767090 -0.312500000    0.249511719
-0.343750000   -0.173767090 -0.328125000    0.032951355
-0.335937500   -0.071623802 -0.328125000    0.032951355
-0.332031250   -0.019641995 -0.328125000    0.032951355
-0.332031250   -0.019641995 -0.330078125    0.006578013
-0.331054688   -0.006551130 -0.330078125    0.006578013
-0.331054688   -0.006551130 -0.330566406    0.000008654
-0.330810547   -0.003272435 -0.330566406    0.000008654
-0.330688477   -0.001632190 -0.330566406    0.000008654
-0.330627441   -0.000811843 -0.330566406    0.000008654
-0.330596924   -0.000401613 -0.330566406    0.000008654
-0.330581665   -0.000196485 -0.330566406    0.000008654
-0.330574036   -0.000093917 -0.330566406    0.000008654
-0.330570221   -0.000042632 -0.330566406    0.000008654
-0.330568314   -0.000016989 -0.330566406    0.000008654
-0.330567360   -0.000004168 -0.330566406    0.000008654

     Next Middle will get us close enough to zero:

     F( -0.330567122 ) is  -0.000000962  

     The desired approximation of the solution is:

       x ≓ -0.330567122

     Note, ≓ is the approximation symbol

One solution was found :