Solution - Finding the roots of polynomials

Step  1  :

            2k + 1
 Simplify   ——————
              3   

Equation at the end of step  1  :

                    (2k + 1)     
  ((k • (2k - 1)) • ————————) +  (2k + 1)2
                       3         

Step  2  :

Equation at the end of step  2  :

                  (2k + 1)     
  (k • (2k - 1) • ————————) +  (2k + 1)2
                     3         

Step  3  :

Equation at the end of step  3  :

  k • (2k - 1) • (2k + 1)    
  ——————————————————————— +  (2k + 1)2
             3               

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  3  as the denominator :

                 (2k + 1)2     (2k + 1)2 • 3
    (2k + 1)2 =  —————————  =  —————————————
                     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 :

 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:

 k • (2k-1) • (2k+1) + (2k+1)2 • 3     4k3 + 12k2 + 11k + 3
 —————————————————————————————————  =  ————————————————————
                 3                              3          

Checking for a perfect cube :

 4.3    4k³ + 12k² + 11k + 3  is not a perfect cube

Trying to factor by pulling out :

 4.4      Factoring:  4k³ + 12k² + 11k + 3 

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

Group 1:  11k + 3 
Group 2:  4k³ + 12k² 

Pull out from each group separately :

Group 1:   (11k + 3) • (1)
Group 2:   (k + 3) • (4k²)

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 :

 4.5    Find roots (zeroes) of :       F(k) = 4k³ + 12k² + 11k + 3
Polynomial Roots Calculator is a set of methods aimed at finding values of  k  for which   F(k)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  k  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  4  and the Trailing Constant is  3.

 The factor(s) are:

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

 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
   4k³ + 12k² + 11k + 3 
can be divided by 3 different polynomials,including by  2k + 3 

Polynomial Long Division :

 4.6    Polynomial Long Division
Dividing :  4k³ + 12k² + 11k + 3 
                              ("Dividend")
By         :    2k + 3    ("Divisor")

Quotient :  2k²+3k+1  Remainder:  0 

Trying to factor by splitting the middle term

 4.7     Factoring  2k²+3k+1 

The first term is,  2k²  its coefficient is  2 .
The middle term is,  +3k  its coefficient is  3 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   2 • 1 = 2 

Step-2 : Find two factors of  2  whose sum equals the coefficient of the middle term, which is   3 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  1  and  2 
                     2k² + 1k + 2k + 1

Step-4 : Add up the first 2 terms, pulling out like factors :
                    k • (2k+1)
              Add up the last 2 terms, pulling out common factors :
                     1 • (2k+1)
Step-5 : Add up the four terms of step 4 :
                    (k+1)  •  (2k+1)
             Which is the desired factorization

Final result :

  (2k + 1) • (k + 1) • (2k + 3)
  —————————————————————————————
                3