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How do you simplify #(x^3y - xy^3)/(xy^2 - x^2 y)#?

Answer 1

Genesis Allen

Assuming #x!=0 and y!=0 and x!=y#

#(x^3y - xy^3)/(xy^2-x^2y)#

#color(white)("xxxxxxxx")#extract common factors

#=((xy)(x^2-y^2))/((xy)(y-x))#

#color(white)("xxxxxxxx")# divide out common factors, factor numerator, & negate denominator

#= ((x+y)(x-y))/((-1)(x-y))#

#color(white)("xxxxxxxx")#divide out common factor

#= -(x+y)#

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Answer 2

Ethan Adkins

To simplify the expression (x^3y - xy^3)/(xy^2 - x^2y), we can factor out the common term "xy" from both the numerator and denominator. This gives us xy(x^2 - y^2)/(xy(y - x)).

Next, we can simplify further by canceling out the common factors. The expression becomes (x^2 - y^2)/(y - x).

Finally, we can factor the numerator using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). Applying this formula, we get (x + y)(x - y)/(y - x).

However, we can further simplify by factoring out a negative sign from the denominator, resulting in (x + y)(y - x)/-(x - y).

Therefore, the simplified form of (x^3y - xy^3)/(xy^2 - x^2y) is -(x + y).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.