Step-by-step explanation:
(3m+2n)/(m-n2)+(5m+n)/(n-m2)-(2m-3n)/(m2-n)
Final result :
-3m3 - 2m2n + 7m2 - 7mn2 + mn + 2n3 + 2n2
—————————————————————————————————————————
(m - n2) • (n - m2)
Step by step solution :
Step 1 :
2m - 3n
Simplify ———————
m2 - n
Trying to factor as a Difference of Squares :
1.1 Factoring: m2 - n
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : m2 is the square of m1
Check : n1 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares
Equation at the end of step 1 :
(3m+2n) (5m+n) (2m-3n)
(————————+————————)-———————
(m-(n2)) (n-(m2)) m2-n
Step 2 :
5m + n
Simplify ——————
n - m2
Trying to factor as a Difference of Squares :
2.1 Factoring: n - m2
Check : n1 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares
Equation at the end of step 2 :
(3m+2n) (5m+n) (2m-3n)
(————————+——————)-———————
(m-(n2)) n-m2 m2-n
Step 3 :
3m + 2n
Simplify ———————
m - n2
Trying to factor as a Difference of Squares :
3.1 Factoring: m - n2
Check : m1 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares
Equation at the end of step 3 :
(3m + 2n) (5m + n) (2m - 3n)
(————————— + ————————) - —————————
m - n2 n - m2 m2 - n
Step 4 :
Calculating the Least Common Multiple :
4.1 Find the Least Common Multiple
The left denominator is : m-n2
The right denominator is : n-m2
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
m-n2 1 0 1
n-m2 0 1 1
Least Common Multiple:
(m-n2) • (n-m2)
Calculating Multipliers :
4.2 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 = n-m2
Right_M = L.C.M / R_Deno = m-n2
