x^3-x^2y over x^2-y^2, divided by x^3y over xy^2+y^3
My solution: y over x
Am I right?
yes
again, what about restrictions?
I'm fairly sure that's the extent to which I need to answer these problems.. I'm not looking for x, therefore restrictions are irrelevant
yes they are relevant
by using the = sign we are saying
"what I write next will be exactly equal to what I had before"
e.g.
(x^2 - 4)/(x-2)
= (x+2)(x-2)/(x-2)
= x+2
for every value of x my last expression x+2 is equal to (x^2 - 4)/(x-2) except when x=2
when x = 2, our first line is 0/0
but our last line is 4
so is 0/0 = 4 ????
that is why whenever we 'cancel' a variable expression we have to restrict those values that make the divided expression equal to zero.
that is ....
(x^2 - 4)/(x-2)
= (x+2)(x-2)/(x-2)
= x+2 , x not equal to 2
To simplify the given expression, we can start by dividing both the numerator and denominator by the greatest common factor.
First, let's simplify the numerator: x^3 - x^2y.
Factor out the common term, which is x^2:
x^3 - x^2y = x^2(x - y)
Next, let's simplify the denominator: x^2 - y^2.
This is a difference of squares, so we can factor it:
x^2 - y^2 = (x - y)(x + y)
Now, let's rewrite the expression with the simplified numerator and denominator:
(x^2(x - y))/((x - y)(x + y))
Next, we can cancel out the common factor, (x - y), in both the numerator and denominator:
(x^2(x - y))/((x - y)(x + y)) = x^2/(x + y)
Now let's simplify the second part of the expression, x^3y/(xy^2 + y^3):
Factor out common term, which is xy:
x^3y/(xy^2 + y^3) = xy(x^2)/(y^2(x + y))
Now, let's rewrite the expression with the simplified numerator and denominator:
(xy(x^2))/(y^2(x + y))
Next, we can cancel out the common factor, y^2, in both the numerator and denominator:
(xy(x^2))/(y^2(x + y)) = x(x^2)/(x + y)
Now, let's divide the two simplified expressions:
(x^2/(x + y)) / (x(x^2)/(x + y))
Dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite the expression as:
(x^2/(x + y)) * ((x + y)/(x(x^2)))
Now we can cancel out the common factor, (x + y):
(x^2 * (x + y))/(x(x^2))
Simplifying the expression further:
(x^3 + xy)/(x^3)
Now, since we need to express the result as y/x, let's switch the numerator and denominator:
(x^3)/(x^3 + xy)
Finally, since the exponent of x in both the numerator and denominator is the same, we can express the result using the rule of exponents (x^n / x^n = 1):
1/(1 + y/x)
This can be further simplified by multiplying the numerator and denominator by x:
x/(x + y)
Therefore, the correct simplified expression is x/(x + y).
Based on your answer of y/x, it seems there may have been an error made during the simplification process.