What is the quotient? 2 m + 4 /8 divided by m + 2/ 6

   12m2 + 2m + 3

 —————————

                6m      

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      

Divide  —  by  m

        2      

Equation at the end of step  3  :

         1     1

 (2m +  ——) +  —

        2m     3

Step  4  :

Rewriting the whole as an Equivalent Fraction :

4.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  2m  as the denominator :

          2m     2m • 2m

    2m =  ——  =  ———————

          1        2m  

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:

2m • 2m + 1     4m2 + 1

———————————  =  ———————

    2m            2m  

Equation at the end of step  4  :

 (4m2 + 1)    1

 ————————— +  —

    2m        3

Step  5  :

Polynomial Roots Calculator :

5.1    Find roots (zeroes) of :       F(m) = 4m2+1

Polynomial Roots Calculator is a set of methods aimed at finding values of  m  for which   F(m)=0  

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 :  1,2 ,4

of the Trailing Constant :  1

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        5.00    

     -1       2        -0.50        2.00    

     -1       4        -0.25        1.25    

     1       1        1.00        5.00    

     1       2        0.50        2.00    

     1       4        0.25        1.25    

Polynomial Roots Calculator found no rational roots

Calculating the Least Common Multiple :

5.2    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    

   Factor      Left

Denominator   Right

Denominator   L.C.M = Max

{Left,Right}

m  101

     Least Common Multiple:

     6m

Calculating Multipliers :

5.3    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      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    

  R. Mult. • R. Num.      2m

  ——————————————————  =   ——

        L.C.M             6m

Adding fractions that have a common denominator :

5.5       Adding up the two equivalent fractions

(4m2+1) • 3 + 2m     12m2 + 2m + 3

————————————————  =  —————————————

       6m                 6m      

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      

Processing ends successfully

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