Solution - Finding the roots of polynomials

Step  1  :

Equation at the end of step  1  :

               12 
  ((((t2)-7t)+————)-(2•3t2))+9
              (t3)
 Step  2  :
            12
 Simplify   ——
            t3

Equation at the end of step  2  :

                    12                  
  ((((t2) -  7t) +  ——) -  (2•3t2)) +  9
                    t3                  

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  t³  as the denominator :

                t2 - 7t     (t2 - 7t) • t3
     t2 - 7t =  ———————  =  ——————————————
                   1              t3      

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

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   t² - 7t  =   t • (t - 7) 

Adding fractions that have a common denominator :

 4.2       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 • (t-7) • t3 + 12     t5 - 7t4 + 12
 ———————————————————  =  —————————————
         t3                   t3      

Equation at the end of step  4  :

   (t5 - 7t4 + 12)                 
  (——————————————— -  (2•3t2)) +  9
         t3                        

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  t³  as the denominator :

               (2•3t2)     (2•3t2) • t3
    (2•3t2) =  ———————  =  ————————————
                  1             t3     

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(t) = t⁵ - 7t⁴ + 12
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  1  and the Trailing Constant is  12.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,12

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (t5-7t4+12) - ((2•3t2) • t3)      -5t5 - 7t4 + 12
 ————————————————————————————  =  ———————————————
              t3                        t3       

Equation at the end of step  5  :

  (-5t5 - 7t4 + 12)    
  ————————————————— +  9
         t3            

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  t³  as the denominator :

         9     9 • t3
    9 =  —  =  ——————
         1       t3  

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -5t⁵ - 7t⁴ + 12  =   -1 • (5t⁵ + 7t⁴ - 12) 

Polynomial Roots Calculator :

 7.2    Find roots (zeroes) of :       F(t) = 5t⁵ + 7t⁴ - 12

     See theory in step 5.2
In this case, the Leading Coefficient is  5  and the Trailing Constant is  -12.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,12

 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
   5t⁵ + 7t⁴ - 12 
can be divided with  t - 1 

Polynomial Long Division :

 7.3    Polynomial Long Division
Dividing :  5t⁵ + 7t⁴ - 12 
                              ("Dividend")
By         :    t - 1    ("Divisor")

Quotient :  5t⁴+12t³+12t²+12t+12  Remainder:  0 

Polynomial Roots Calculator :

 7.4    Find roots (zeroes) of :       F(t) = 5t⁴+12t³+12t²+12t+12

     See theory in step 5.2
In this case, the Leading Coefficient is  5  and the Trailing Constant is  12.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,12

 Let us test ....

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

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 (-5t4-12t3-12t2-12t-12) • (t-1) + 9 • t3      -5t5 - 7t4 + 9t3 + 12
 ————————————————————————————————————————  =  —————————————————————
                    t3                                 t3          

Checking for a perfect cube :

 7.6    -5t⁵ - 7t⁴ + 9t³ + 12  is not a perfect cube

Trying to factor by pulling out :

 7.7      Factoring:  -5t⁵ - 7t⁴ + 9t³ + 12 

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

Group 1:  9t³ + 12 
Group 2:  -5t⁵ - 7t⁴ 

Pull out from each group separately :

Group 1:   (3t³ + 4) • (3)
Group 2:   (5t + 7) • (-t⁴)

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 :

 7.8    Find roots (zeroes) of :       F(t) = -5t⁵ - 7t⁴ + 9t³ + 12

     See theory in step 5.2
In this case, the Leading Coefficient is  -5  and the Trailing Constant is  12.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,12

 Let us test ....

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

Polynomial Roots Calculator found no rational roots

Final result :

  5t5 - 7t4 + 9t3 + 12
  —————————————————————
           t3