Final result :
(b - a) • (a2 + ab + b2)
————————————————————————
a2b3
Step by step solution :
Step 1 :
1
Simplify —
a
Equation at the end of step 1 :
1 1 1
————-———— ÷ (—•b)
(a2) (b2) a
Step 2 :
1
Simplify ——
b2
Equation at the end of step 2 :
1 1 b
———— - —— ÷ —
(a2) b2 a
Step 3 :
1 b
Divide —— by —
b2 a
3.1 Dividing fractions
To divide fractions, write the divison as multiplication by the reciprocal of the divisor :
1 b 1 a
—— ÷ — = —— • —
b2 a b2 b
Multiplying exponential expressions :
3.2 b2 multiplied by b1 = b(2 + 1) = b3
Equation at the end of step 3 :
1 a
———— - ——
(a2) b3
Step 4 :
1
Simplify ——
a2
Equation at the end of step 4 :
1 a
—— - ——
a2 b3
Step 5 :
Calculating the Least Common Multiple :
5.1 Find the Least Common Multiple
The left denominator is : a2
The right denominator is : b3
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
a 2 0 2
b 0 3 3
Least Common Multiple:
a2b3
Calculating Multipliers :
5.2 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 = b3
Right_M = L.C.M / R_Deno = a2
Making Equivalent Fractions :
5.3 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. b3
—————————————————— = ————
L.C.M a2b3
R. Mult. • R. Num. a • a2
—————————————————— = ——————
L.C.M a2b3
Adding fractions that have a common denominator :
5.4 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:
b3 - (a • a2) b3 - a3
————————————— = ———————
a2b3 a2b3
Trying to factor as a Difference of Cubes:
5.5 Factoring: b3 - a3
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0+b3 =
a3+b3
Check : b3 is the cube of b1
Check : a3 is the cube of a1
Factorization is :
(b - a) • (b2 + ab + a2)
Trying to factor a multi variable polynomial :
5.6 Factoring b2 + ab + a2
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Final result :
(b - a) • (a2 + ab + b2)
————————————————————————
a2b3
