If $ x = 5 $, $ y = 3 $, the value of $ \frac{x^3 - y^3}{x^2 - y^2} - \frac{3xy}{x + y} $ will be

### Concept & Formula This problem tests your ability to simplify rational algebraic expressions before inserting numerical values. The goal is to combine the two fractions into a much simpler form. The necessary factorization formulas are: $$ x^3 - y^3 = (x - y)(x^2 + xy + y^2) $$ $$ x^2 - y^2 = (x - y)(x + y) $$ ### Step-by-Step Solution Let's start with the first fraction in the expression: $$ \frac{x^3 - y^3}{x^2 - y^2} $$ Factor both the numerator and the denominator using the standard formulas: $$ = \frac{(x - y)(x^2 + xy + y^2)}{(x - y)(x + y)} $$ Cancel out the common $ (x - y) $ term from the top and bottom: $$ = \frac{x^2 + xy + y^2}{x + y} $$ Now, bring back the second part of the original expression and subtract it: $$ \frac{x^2 + xy + y^2}{x + y} - \frac{3xy}{x + y} $$ Because both fractions now share a common denominator ($ x + y $), we can combine their numerators: $$ = \frac{x^2 + xy + y^2 - 3xy}{x + y} $$ Combine the like terms ($ +xy - 3xy = -2xy $): $$ = \frac{x^2 - 2xy + y^2}{x + y} $$ Notice that the numerator is now a perfect square trinomial. Factor it: $$ = \frac{(x - y)^2}{x + y} $$ The expression is fully simplified. Now substitute the given values: $ x = 5 $ and $ y = 3 $. $$ = \frac{(5 - 3)^2}{5 + 3} $$ $$ = \frac{(2)^2}{8} $$ $$ = \frac{4}{8} $$ $$ = \frac{1}{2} $$ ### Exam Strategy & Shortcut Instead of algebraic manipulation, you can use direct substitution ("Value Putting") immediately, as the numbers are very small. Fraction 1: $ \frac{5^3 - 3^3}{5^2 - 3^2} = \frac{125 - 27}{25 - 9} = \frac{98}{16} = \frac{49}{8} $ Fraction 2: $ \frac{3(5)(3)}{5 + 3} = \frac{45}{8} $ Difference: $ \frac{49}{8} - \frac{45}{8} = \frac{4}{8} = \frac{1}{2} $. For small integers, basic arithmetic is often faster than performing multi-step polynomial factorization. ### Common Pitfall When simplifying algebraically, students sometimes mistakenly cancel $ (x^2 - y^2) $ out of the denominator without properly factoring the numerator, or miscalculate the combination $ +xy - 3xy $ as $ +2xy $, which leads to an entirely incorrect final value. ### Final Answer **Therefore, the correct answer is $ \frac{1}{2} $.**