Solution - Finding the roots of polynomials

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.001" was replaced by "(001/1000)".

Step  1  :

              1 
 Simplify   ————
            1000

Equation at the end of step  1  :

                  1 
  ((t + 9)3) -  ————
                1000

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  1000  as the denominator :

                 (t + 9)3     (t + 9)3 • 1000
     (t + 9)3 =  ————————  =  ———————————————
                    1              1000      

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:

 (t+9)3 • 1000 - (1)     1000t3 + 27000t2 + 243000t + 728999
 ———————————————————  =  ———————————————————————————————————
        1000                            1000                

Checking for a perfect cube :

 2.3    1000t³ + 27000t² + 243000t + 728999  is not a perfect cube

Trying to factor by pulling out :

 2.4      Factoring:  1000t³ + 27000t² + 243000t + 728999 

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

Group 1:  1000t³ + 728999 
Group 2:  27000t² + 243000t 

Pull out from each group separately :

Group 1:   (1000t³ + 728999) • (1)
Group 2:   (t + 9) • (27000t)

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 :

 2.5    Find roots (zeroes) of :       F(t) = 1000t³ + 27000t² + 243000t + 728999
Polynomial Roots Calculator is a set of methods aimed at finding values of  t  for which   F(t)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  t  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  1000  and the Trailing Constant is  728999.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4 ,5 ,8 ,10 ,20 ,25 ,40 ,50 , etc
 of the Trailing Constant :  1 ,89 ,8191 ,728999

 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   1000t³ + 27000t² + 243000t + 728999 
can be divided with  10t + 89 

Polynomial Long Division :

 2.6    Polynomial Long Division
Dividing :  1000t³ + 27000t² + 243000t + 728999 
                              ("Dividend")
By         :    10t + 89    ("Divisor")

Quotient :  100t²+1810t+8191  Remainder:  0 

Trying to factor by splitting the middle term

 2.7     Factoring  100t²+1810t+8191 

The first term is,  100t²  its coefficient is  100 .
The middle term is,  +1810t  its coefficient is  1810 .
The last term, "the constant", is  +8191 

Step-1 : Multiply the coefficient of the first term by the constant   100 • 8191 = 819100 

Step-2 : Find two factors of  819100  whose sum equals the coefficient of the middle term, which is   1810 .

For tidiness, printing of 30 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  (100t2 + 1810t + 8191) • (10t + 89)
  ———————————————————————————————————
                 1000