Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Step 1 :
Equation at the end of step 1 :
((4•(a2))-1) ((4•(b2))-1) ((4•(c2))-1) (—————————————+—————————————)+———————————— ((a-b)•(a-c)) ((b-c)•(b-a)) (c-a)•(c-b)Step 2 :
Equation at the end of step 2 :
((4•(a2))-1) ((4•(b2))-1) (22c2-1)
(—————————————+—————————————)+———————————
((a-b)•(a-c)) ((b-c)•(b-a)) (c-a)•(c-b)
Step 3 :
4c2 - 1
Simplify —————————————————
(c - a) • (c - b)
Trying to factor as a Difference of Squares :
3.1 Factoring: 4c2 - 1
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 : 4 is the square of 2
Check : 1 is the square of 1
Check : c2 is the square of c1
Factorization is : (2c + 1) • (2c - 1)
Equation at the end of step 3 :
((4•(a2))-1) ((4•(b2))-1) (2c+1)•(2c-1)
(—————————————+—————————————)+—————————————
((a-b)•(a-c)) ((b-c)•(b-a)) (c-a)•(c-b)
Step 4 :
Equation at the end of step 4 :
((4•(a2))-1) ((4•(b2))-1) (2c+1)•(2c-1) (—————————————+————————————)+————————————— ((a-b)•(a-c)) (b-c)•(b-a) (c-a)•(c-b)Step 5 :
Equation at the end of step 5 :
((4•(a2))-1) (22b2-1) (2c+1)•(2c-1)
(—————————————+———————————)+—————————————
((a-b)•(a-c)) (b-c)•(b-a) (c-a)•(c-b)
Step 6 :
4b2 - 1
Simplify —————————————————
(b - c) • (b - a)
Trying to factor as a Difference of Squares :
6.1 Factoring: 4b2 - 1
Check : 4 is the square of 2
Check : 1 is the square of 1
Check : b2 is the square of b1
Factorization is : (2b + 1) • (2b - 1)
Equation at the end of step 6 :
((4•(a2))-1) (2b+1)•(2b-1) (2c+1)•(2c-1)
(—————————————+—————————————)+—————————————
((a-b)•(a-c)) (b-c)•(b-a) (c-a)•(c-b)
Step 7 :
Equation at the end of step 7 :
((4•(a2))-1) (2b+1)•(2b-1) (2c+1)•(2c-1) (————————————+—————————————)+————————————— (a-b)•(a-c) (b-c)•(b-a) (c-a)•(c-b)Step 8 :
Equation at the end of step 8 :
(22a2-1) (2b+1)•(2b-1) (2c+1)•(2c-1)
(———————————+—————————————)+—————————————
(a-b)•(a-c) (b-c)•(b-a) (c-a)•(c-b)
Step 9 :
4a2 - 1
Simplify —————————————————
(a - b) • (a - c)
Trying to factor as a Difference of Squares :
9.1 Factoring: 4a2 - 1
Check : 4 is the square of 2
Check : 1 is the square of 1
Check : a2 is the square of a1
Factorization is : (2a + 1) • (2a - 1)
Equation at the end of step 9 :
(2a+1)•(2a-1) (2b+1)•(2b-1) (2c+1)•(2c-1)
(—————————————+—————————————)+—————————————
(a-b)•(a-c) (b-c)•(b-a) (c-a)•(c-b)
Step 10 :
Calculating the Least Common Multiple :
10.1 Find the Least Common Multiple
The left denominator is : (a-b) • (a-c)
The right denominator is : (b-c) • (b-a)
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| a-b | 1 | 1 | 1 |
| a-c | 1 | 0 | 1 |
| b-c | 0 | 1 | 1 |
Least Common Multiple:
(a-b) • (a-c) • (b-c)
Calculating Multipliers :
10.2 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 = b-c
Right_M = L.C.M / R_Deno = -1•(a-c)
Making Equivalent Fractions :
10.3 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. (2a+1) • (2a-1) • (b-c) —————————————————— = ——————————————————————— L.C.M (a-b) • (a-c) • (b-c) R. Mult. • R. Num. (2b+1) • (2b-1) • -1 • (a-c) —————————————————— = ———————————————————————————— L.C.M (a-b) • (a-c) • (b-c)
Adding fractions that have a common denominator :
10.4 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:
(2a+1) • (2a-1) • (b-c) + (2b+1) • (2b-1) • -1 • (a-c) 4a2b-4a2c-4ab2+a+4b2c-b
—————————————————————————————————————————————————————— = ———————————————————————
(a-b) • (a-c) • (b-c) (a-b) • (a-c) • (b-c)
Equation at the end of step 10 :
(4a2b-4a2c-4ab2+a+4b2c-b) (2c+1)•(2c-1)
—————————————————————————+—————————————
(a-b)•(a-c)•(b-c) (c-a)•(c-b)
Step 11 :
Trying to factor by pulling out :
11.1 Factoring: 4a2b-4a2c-4ab2+a+4b2c-b
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -4ab2+4b2c
Group 2: 4a2b-4a2c
Group 3: a-b
Pull out from each group separately :
Group 1: (a-c) • (-4b2)
Group 2: (b-c) • (4a2)
Group 3: (a-b) • (1)
Looking for common sub-expressions :
Group 1: (a-c) • (-4b2)
Group 3: (a-b) • (1)
Group 2: (b-c) • (4a2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Calculating the Least Common Multiple :
11.2 Find the Least Common Multiple
The left denominator is : (a-b) • (a-c) • (b-c)
The right denominator is : (c-a) • (c-b)
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| a-b | 1 | 0 | 1 |
| a-c | 1 | 1 | 1 |
| b-c | 1 | 1 | 1 |
Least Common Multiple:
(a-b) • (a-c) • (b-c)
Calculating Multipliers :
11.3 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 = 1
Right_M = L.C.M / R_Deno = (a-b)•-1•-1
Making Equivalent Fractions :
11.4 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. (4a2b-4a2c-4ab2+a+4b2c-b) —————————————————— = ————————————————————————— L.C.M (a-b) • (a-c) • (b-c) R. Mult. • R. Num. (2c+1) • (2c-1) • (a-b) • -1 • -1 —————————————————— = ————————————————————————————————— L.C.M (a-b) • (a-c) • (b-c)
Adding fractions that have a common denominator :
11.5 Adding up the two equivalent fractions
(4a2b-4a2c-4ab2+a+4b2c-b) + (2c+1) • (2c-1) • (a-b) • -1 • -1 4a2b-4a2c-4ab2+4ac2+4b2c-4bc2
————————————————————————————————————————————————————————————— = —————————————————————————————
(a-b) • (a-c) • (b-c) (a-b) • (a-c) • (b-c)
Step 12 :
Pulling out like terms :
12.1 Pull out like factors :
4a2b - 4a2c - 4ab2 + 4ac2 + 4b2c - 4bc2 =
4 • (a2b - a2c - ab2 + ac2 + b2c - bc2)
Trying to factor by pulling out :
12.2 Factoring: a2b - a2c - ab2 + ac2 + b2c - bc2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: b2c - ab2
Group 2: a2b - a2c
Group 3: ac2 - bc2
Pull out from each group separately :
Group 1: (a - c) • (-b2)
Group 2: (b - c) • (a2)
Group 3: (a - b) • (c2)
Looking for common sub-expressions :
Group 1: (a - c) • (-b2)
Group 3: (a - b) • (c2)
Group 2: (b - c) • (a2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Final result :
4 • (a2b + a2c + ab2 + ac2 + b2c + bc2)
———————————————————————————————————————
(a + b) • (a + c) • (b + c)
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