Answer: (5x2 + 3y2) • (2x2 + y2) ———————————————————————— x2y2
Explanation:
STEP 1: y2
Simplify ——
x2
Equation at the end of step 1: (x2) y2
((10•————)-1)-(3•——)
(y2) x2
STEP 2 : x2
Simplify ——
y2
Equation at the end of step 2: x2 3y2
((10 • ——) - 1) - ———
y2 x2
STEP 3:Rewriting the whole as an Equivalent Fraction 3.1 Subtracting a whole from a fraction Rewrite the whole as a fraction using y2 as the denominator : 1 1 • y2
1 = — = ——————
1 y2
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 denominatorAdding fractions that have a common denominator : 3.2 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: 10x2 - (y2) 10x2 - y2
——————————— = —————————
y2 y2
Equation at the end of step 3: (10x2 - y2) 3y2
——————————— - ———
y2 x2
STEP 4:Trying to factor as a Difference of Squares: 4.1 Factoring: 10x2-y2 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 - B2Note : AB = BA is the commutative property of multiplication. Note : - AB + AB equals zero and is therefore eliminated from the expression.Check : 10 is not a square !! Ruling : Binomial can not be factored as thedifference of two perfect squaresCalculating the Least Common Multiple : 4.2 Find the Least Common Multiple The left denominator is : y2 The right denominator is : x2 Number of times each Algebraic Factor appears in the factorization of: Algebraic Factor Left Denominator Right Denominator L.C.M = Max {Left,Right} x 022 y 202 Least Common Multiple: x2y2 Calculating Multipliers : 4.3 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 = x2 Right_M = L.C.M / R_Deno = y2Making Equivalent Fractions : 4.4 Rewrite the two fractions into equivalent fractionsTwo fractions are called equivalent if they have the same numeric value.