Reduce each rational expression to lowest terms:
3x^2-12y^2 over x^2+4xy+4y^2
(3x^2 - 12y^2) / (x^2 + 4xy + 4y^2),
3(x^2 - 4y^2) / (x + 2y) (x + 2y),
3(x + 2y) (x - 2y) / (x + 2y)(x - 2y),
Cancel two (x + 2y) factors and get:
3(x - 2y) / (x + 2y).
CORRECTION: The denominator of the 3rd expression should be (x + 2y) (x + 2y).
To reduce the rational expression (3x^2-12y^2) / (x^2+4xy+4y^2) to the lowest terms, we need to factor both the numerator and denominator and cancel out any common factors.
First, let's factor the numerator and denominator.
The numerator is a difference of squares, so we can apply the formula (a^2 - b^2) = (a + b)(a - b):
3x^2 - 12y^2 = 3(x^2 - 4y^2) = 3(x + 2y)(x - 2y)
The denominator is a perfect square trinomial, so we can rewrite it as a square of a binomial:
x^2 + 4xy + 4y^2 = (x + 2y)^2
Now, we can cancel out the common factor of (x + 2y) in the numerator and denominator:
(3(x + 2y)(x - 2y)) / ((x + 2y)^2)
After canceling out the common factor, the reduced expression is:
3(x - 2y) / (x + 2y)
To reduce a rational expression to its lowest terms, you need to simplify both the numerator and denominator as much as possible. In this case, we have the rational expression:
(3x^2 - 12y^2) / (x^2 + 4xy + 4y^2)
To simplify this expression, start by factoring the numerator and denominator.
First, factor out the greatest common factor in the numerator, which is 3:
3(x^2 - 4y^2)
The numerator can be further factored using the difference of squares formula: a^2 - b^2 = (a + b)(a - b):
3(x + 2y)(x - 2y)
Now let's factor the denominator:
x^2 + 4xy + 4y^2
This expression is a perfect square trinomial and can be factored as:
(x + 2y)^2
Now, we can rewrite the rational expression in factored form:
3(x + 2y)(x - 2y) / (x + 2y)^2
Notice that the factor (x + 2y) appears in both the numerator and the denominator. You can cancel out this common factor:
3(x - 2y) / (x + 2y)
This is the reduced form of the rational expression.