Simplify the rational expression
(x^2+xy-2y^2)/(x^2+3xy+2y^2)
did you try factoring both expressions?
They factor quite easily.
Then divide out the common factor, don't forget to state the restriction.
But it has two variables?
Simplify the rational expression
(x^2+xy-2y^2)/(x^2+3xy+2y^2)
Algebra 2 - Damon, Thursday, March 27, 2014 at 11:04am
[ (x-y)(x+2y) ] / [(x+2y)(x+y) ]
= (x-y)/(x+y)
To simplify the rational expression ((x^2+xy-2y^2)/(x^2+3xy+2y^2)), we can factor the numerator and denominator and then cancel out any common factors.
Let's start with factoring the numerator and the denominator:
Numerator: x^2 + xy - 2y^2
To factor this quadratic expression, we need to find two numbers whose sum is the coefficient of the middle term (xy) and whose product is equal to the product of the coefficient of the first term (x^2) and the coefficient of the last term (-2y^2).
The coefficients are:
a = 1
b = 1
c = -2
So, we need to find two numbers whose sum is 1 and whose product is -2. These numbers are 2 and -1 because 2 + (-1) = 1 and 2 * (-1) = -2.
Using these numbers, we can rewrite the numerator as:
x^2 + 2xy - yx - 2y^2
Now, we group the terms:
(x^2 + 2xy) + (-yx - 2y^2)
Next, we factor out the common factors from each group:
x(x + 2y) - y(x + 2y)
We can see that we have a common factor of (x + 2y) in both terms, so we can factor it out:
(x - y)(x + 2y)
Therefore, the numerator is factored as: (x - y)(x + 2y)
Denominator: x^2 + 3xy + 2y^2
Similarly, we can factor the denominator by finding two numbers whose sum is the coefficient of the middle term (3xy) and whose product is equal to the product of the coefficient of the first term (x^2) and the coefficient of the last term (2y^2).
The coefficients are:
a = 1
b = 3
c = 2
So, we need to find two numbers whose sum is 3 and whose product is 2. These numbers are 2 and 1 because 2 + 1 = 3 and 2 * 1 = 2.
Using these numbers, we can rewrite the denominator as:
x^2 + 2xy + xy + 2y^2
Now, we group the terms:
(x^2 + 2xy) + (xy + 2y^2)
Next, we factor out the common factors from each group:
x(x + 2y) + y(x + 2y)
We can see that we have a common factor of (x + 2y) in both terms, so we can factor it out:
(x + 2y)(x + y)
Therefore, the denominator is factored as: (x + 2y)(x + y)
Now that we have factored the numerator and the denominator, we can simplify the rational expression:
((x^2+xy-2y^2)/(x^2+3xy+2y^2)) = (x - y)(x + 2y)/(x + 2y)(x + y)
Notice that the (x + 2y) terms in the numerator and the denominator cancel out, leaving us with:
(x - y)/(x + y)
So, the simplified form of the rational expression (x^2+xy-2y^2)/(x^2+3xy+2y^2) is (x - y)/(x + y).