Solution - Finding the roots of polynomials

Step  1  :

            12
 Simplify   ——
            a2

Equation at the end of step  1  :

                     12            
  ((((a2) +  11a) -  ——) -  2a) +  3
                     a2            

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  a²  as the denominator :

                 a2 + 11a     (a2 + 11a) • a2
     a2 + 11a =  ————————  =  ———————————————
                    1               a2       

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  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   a² + 11a  =   a • (a + 11) 

Adding fractions that have a common denominator :

 3.2       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:

 a • (a+11) • a2 - (12)     a4 + 11a3 - 12
 ——————————————————————  =  ——————————————
           a2                     a2      

Equation at the end of step  3  :

   (a4 + 11a3 - 12)           
  (———————————————— -  2a) +  3
          a2                  

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  a²  as the denominator :

          2a     2a • a2
    2a =  ——  =  ———————
          1        a2   

Polynomial Roots Calculator :

 4.2    Find roots (zeroes) of :       F(a) = a⁴ + 11a³ - 12
Polynomial Roots Calculator is a set of methods aimed at finding values of  a  for which   F(a)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  a  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 ....

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
   a⁴ + 11a³ - 12 
can be divided with  a - 1 

Polynomial Long Division :

 4.3    Polynomial Long Division
Dividing :  a⁴ + 11a³ - 12 
                              ("Dividend")
By         :    a - 1    ("Divisor")

Quotient :  a³+12a²+12a+12  Remainder:  0 

Polynomial Roots Calculator :

 4.4    Find roots (zeroes) of :       F(a) = a³+12a²+12a+12

     See theory in step 4.2
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 :

 4.5       Adding up the two equivalent fractions

 (a3+12a2+12a+12) • (a-1) - (2a • a2)     a4 + 9a3 - 12
 ————————————————————————————————————  =  —————————————
                  a2                           a2      

Equation at the end of step  4  :

  (a4 + 9a3 - 12)    
  ——————————————— +  3
        a2           

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  a²  as the denominator :

         3     3 • a2
    3 =  —  =  ——————
         1       a2  

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(a) = a⁴ + 9a³ - 12

     See theory in step 4.2
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

 (a4+9a3-12) + 3 • a2     a4 + 9a3 + 3a2 - 12
 ————————————————————  =  ———————————————————
          a2                      a2         

Checking for a perfect cube :

 5.4    a⁴ + 9a³ + 3a² - 12  is not a perfect cube

Trying to factor by pulling out :

 5.5      Factoring:  a⁴ + 9a³ + 3a² - 12 

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

Group 1:  3a² - 12 
Group 2:  9a³ + a⁴ 

Pull out from each group separately :

Group 1:   (a² - 4) • (3)
Group 2:   (a + 9) • (a³)

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 :

 5.6    Find roots (zeroes) of :       F(a) = a⁴ + 9a³ + 3a² - 12

     See theory in step 4.2
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

Final result :

  a4 + 9a3 + 3a2 - 12
  ———————————————————
          a2