Adding, Subtracting And Finding The Least Common Multiple

Step 1 :

Equation at the end of step 1 :

(y2) y (y2) (((((x2)+4xy)-((((12•————)•x)•—)•(x3)))-(8x•————))+7xy)+(2•3y2) 4 2 (x2)

Step 2 :

y2 Simplify —— x2

Equation at the end of step 2 :

(y2) y y2 (((((x2)+4xy)-((((12•————)•x)•—)•(x3)))-(8x•——))+7xy)+(2•3y2) 4 2 x2

Step 3 :

y Simplify — 2

Equation at the end of step 3 :

(y2) y 8y2 (((((x2)+4xy)-((((12•————)•x)•—)•x3))-———)+7xy)+(2•3y2) 4 2 x

Step 4 :

y2 Simplify —— 4

Equation at the end of step 4 :

y2 y 8y2 (((((x2)+4xy)-((((12•——)•x)•—)•x3))-———)+7xy)+(2•3y2) 4 2 x

Step 5 :

Multiplying exponential expressions :

5.1 x1 multiplied by x3 = x(1 + 3) = x4

Equation at the end of step 5 :

3x4y3 8y2 (((((x2)+4xy)-—————)-———)+7xy)+(2•3y2) 2 x

Step 6 :

Rewriting the whole as an Equivalent Fraction :

6.1 Subtracting a fraction from a whole Rewrite the whole as a fraction using 2 as the denominator :

x2 + 4xy (x2 + 4xy) • 2 x2 + 4xy = ———————— = —————————————— 1 2

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

Step 7 :

Pulling out like terms :

7.1 Pull out like factors : x2 + 4xy = x • (x + 4y)

Adding fractions that have a common denominator :

7.2 Adding up the two equivalent fractions Add the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • (x+4y) • 2 - (3x4y3) -3x4y3 + 2x2 + 8xy ———————————————————————— = —————————————————— 2 2

Equation at the end of step 7 :

(-3x4y3 + 2x2 + 8xy) 8y2 ((———————————————————— - ———) + 7xy) + (2•3y2) 2 x

Step 8 :

Step 9 :

Pulling out like terms :

9.1 Pull out like factors : -3x4y3 + 2x2 + 8xy = -x • (3x3y3 - 2x - 8y)

Trying to factor a multi variable polynomial :

9.2 Factoring 3x3y3 - 2x - 8y Try to factor this multi-variable trinomial using trial and errorFactorization fails

Calculating the Least Common Multiple :

9.3 Find the Least Common Multiple The left denominator is : 2 The right denominator is : x

Number of times each prime factor appears in the factorization of:

Prime Factor	Left Denominator	Right Denominator	L.C.M = Max {Left,Right}
2	1	0	1
Product of all Prime Factors	2	1	2

Number of times each Algebraic Factor appears in the factorization of:

Algebraic Factor	Left Denominator	Right Denominator	L.C.M = Max {Left,Right}
x	0	1	1

Least Common Multiple: 2x

Calculating Multipliers :

9.4 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 = x Right_M = L.C.M / R_Deno = 2

Making Equivalent Fractions :

9.5 Rewrite the two fractions into equivalent fractionsTwo 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. -x • (3x3y3-2x-8y) • x —————————————————— = —————————————————————— L.C.M 2x R. Mult. • R. Num. 8y2 • 2 —————————————————— = ——————— L.C.M 2x

Adding fractions that have a common denominator :

9.6 Adding up the two equivalent fractions

-x • (3x3y3-2x-8y) • x - (8y2 • 2) -3x5y3 + 2x3 + 8x2y - 16y2 —————————————————————————————————— = —————————————————————————— 2x 2x

Equation at the end of step 9 :

(-3x5y3 + 2x3 + 8x2y - 16y2) (———————————————————————————— + 7xy) + (2•3y2) 2x

Step 10 :

Rewriting the whole as an Equivalent Fraction :

10.1 Adding a whole to a fraction Rewrite the whole as a fraction using 2x as the denominator :

7xy 7xy • 2x 7xy = ——— = ———————— 1 2x

Checking for a perfect cube :

10.2 -3x5y3 + 2x3 + 8x2y - 16y2 is not a perfect cube

Adding fractions that have a common denominator :

10.3 Adding up the two equivalent fractions

(-3x5y3+2x3+8x2y-16y2) + 7xy • 2x -3x5y3 + 2x3 + 22x2y - 16y2 ————————————————————————————————— = ——————————————————————————— 2x 2x

Equation at the end of step 10 :

(-3x5y3 + 2x3 + 22x2y - 16y2) ————————————————————————————— + (2•3y2) 2x

Step 11 :

Rewriting the whole as an Equivalent Fraction :

11.1 Adding a whole to a fraction Rewrite the whole as a fraction using 2x as the denominator :

(2•3y2) (2•3y2) • 2x (2•3y2) = ——————— = ———————————— 1 2x

Checking for a perfect cube :

11.2 -3x5y3 + 2x3 + 22x2y - 16y2 is not a perfect cube

Adding fractions that have a common denominator :

11.3 Adding up the two equivalent fractions

(-3x5y3+2x3+22x2y-16y2) + (2•3y2) • 2x -3x5y3 + 2x3 + 22x2y + 12xy2 - 16y2 —————————————————————————————————————— = ——————————————————————————————————— 2x 2x

Final result :

-3x5y3 + 2x3 + 22x2y + 12xy2 - 16y2 ——————————————————————————————————— 2x