12m2 + 2m + 3
 —————————
        6m   Â
Step-by-step explanation:
Step  1  :
      1
Simplify  —
      3
Equation at the end of step  1  :
    4     1
 (2m +  — ÷ m) + —
    8     3
Step  2  :
      1
Simplify  —
      2
Equation at the end of step  2  :
    1     1
 (2m +  — ÷ m) +  —
    2     3
Step  3  :
    1   Â
Divide  —  by  m
    2   Â
Equation at the end of step  3  :
     1   1
 (2m +  ——) +  —
    2m   3
Step  4  :
Rewriting the whole as an Equivalent Fraction :
4.1 Â Adding a fraction to a whole
Rewrite the whole as a fraction using  2m  as the denominator :
     2m   2m • 2m
  2m =  ——  =  ———————
     1     2m Â
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 Â Â Â 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:
2m • 2m + 1   4m2 + 1
———————————  =  ———————
  2m       2m Â
Equation at the end of step  4  :
 (4m2 + 1)   1
 ————————— +  —
  2m     3
Step  5  :
Polynomial Roots Calculator :
5.1 Â Â Find roots (zeroes) of : Â Â Â F(m) = 4m2+1
Polynomial Roots Calculator is a set of methods aimed at finding values of  m  for which  F(m)=0 Â
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  m  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  1.
The factor(s) are:
of the Leading Coefficient : Â 1,2 ,4
of the Trailing Constant : Â 1
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     5.00  Â
   -1    2     -0.50     2.00  Â
   -1    4     -0.25     1.25  Â
   1    1     1.00     5.00  Â
   1    2     0.50     2.00  Â
   1    4     0.25     1.25  Â
Polynomial Roots Calculator found no rational roots
Calculating the Least Common Multiple :
5.2 Â Â Find the Least Common Multiple
   The left denominator is :    2m
   The right denominator is :    3
    Number of times each prime factor
    appears in the factorization of:
Prime
Factor  Left
Denominator  Right
Denominator  L.C.M = Max
{Left,Right}
2101
3011
Product of all
Prime Factors  236
         Number of times each Algebraic Factor
      appears in the factorization of:
  Algebraic  Â
  Factor    Left
Denominator  Right
Denominator  L.C.M = Max
{Left,Right}
m  101
   Least Common Multiple:
   6m
Calculating Multipliers :
5.3 Â Â Calculate multipliers for the two fractions
  Denote the Least Common Multiple by  L.C.M
  Denote the Left Multiplier by  Left_M
  Denote the Right Multiplier by  Right_M
  Denote the Left Deniminator by  L_Deno
  Denote the Right Multiplier by  R_Deno
 Left_M = L.C.M / L_Deno = 3
 Right_M = L.C.M / R_Deno = 2m
Making Equivalent Fractions :
5.4 Â Â Â Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example :  1/2  and  2/4  are equivalent,  y/(y+1)2  and  (y2+y)/(y+1)3  are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
 L. Mult. • L. Num.    (4m2+1) • 3
 ——————————————————  =  ———————————
    L.C.M         6m  Â
 R. Mult. • R. Num.    2m
 ——————————————————  =  ——
    L.C.M       6m
Adding fractions that have a common denominator :
5.5 Â Â Â Adding up the two equivalent fractions
(4m2+1) • 3 + 2m   12m2 + 2m + 3
————————————————  =  —————————————
    6m         6m   Â
Trying to factor by splitting the middle term
5.6   Factoring  12m2 + 2m + 3
The first term is,  12m2  its coefficient is  12 .
The middle term is,  +2m  its coefficient is  2 .
The last term, "the constant", is  +3
Step-1 : Multiply the coefficient of the first term by the constant  12 • 3 = 36
Step-2 : Find two factors of  36  whose sum equals the coefficient of the middle term, which is  2 .
   -36   +   -1   =   -37
   -18   +   -2   =   -20
   -12   +   -3   =   -15
   -9   +   -4   =   -13
   -6   +   -6   =   -12
   -4   +   -9   =   -13
For tidiness, printing of 12 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 :
 12m2 + 2m + 3
 —————————————
   6m   Â
Processing ends successfully
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