a triangle has side lengths of 1 over x-1, -1 over x^2-1, and 1 over x^2-2x-3 meters. find a simplified expression for the perimeter of the triangle in terms of x. (perimeter is sum of all side lengths)
To find the perimeter of the triangle, we need to add up all the side lengths.
First, let's simplify each side length:
Side 1: 1/(x-1)
Side 2: -1/(x^2-1) = -1/((x-1)(x+1))
Side 3: 1/(x^2-2x-3) = 1/((x+1)(x-3))
Now, let's add up all the side lengths to find the perimeter:
Perimeter = Side 1 + Side 2 + Side 3
= 1/(x-1) + (-1/((x-1)(x+1))) + 1/((x+1)(x-3))
To simplify this expression, we can find a common denominator and add up the fractions:
Perimeter = ((x+1)(x-3) + (-1)(x-1) + (x-1))/((x+1)(x-1)(x-3))
= (x^2 - 2x - 3 - x + 1 + x - 1)/((x+1)(x-1)(x-3))
= (x^2 - 2x - 3)/((x+1)(x-1)(x-3))
Therefore, the simplified expression for the perimeter of the triangle in terms of x is:
Perimeter = (x^2 - 2x - 3)/((x+1)(x-1)(x-3)) meters.
To find the perimeter of the triangle, we need to add the lengths of all three sides.
Given the side lengths of the triangle:
1. Side 1: 1 / (x - 1) meters
2. Side 2: -1 / (x^2 - 1) meters
3. Side 3: 1 / (x^2 - 2x - 3) meters
To simplify the expression, we'll start by finding a common denominator for the three fractions.
The common denominator is (x - 1)(x + 1)(x - 3). Multiplying each side length by the appropriate factor will result in an equivalent fraction:
1. Side 1: (1 / (x - 1)) * ((x + 1)(x - 3) / (x + 1)(x - 3)) = (x + 1)(x - 3) / (x^2 - 1)
2. Side 2: (-1 / (x^2 - 1)) * ((x - 1)(x - 3) / (x - 1)(x - 3)) = -(x - 1)(x - 3) / (x^4 - 2x^3 - 4x^2 + 8x - 6)
3. Side 3: (1 / (x^2 - 2x - 3)) * ((x - 1)(x + 1) / (x - 1)(x + 1)) = (x - 1)(x + 1) / (x^2 - 2x - 3)
Now, we can compute the sum of the three sides:
Perimeter = (x + 1)(x - 3) / (x^2 - 1) - (x - 1)(x - 3) / (x^4 - 2x^3 - 4x^2 + 8x - 6) + (x - 1)(x + 1) / (x^2 - 2x - 3)
To further simplify the expression, we can start by finding a common denominator for the three fractions in the numerator:
Perimeter = [(x + 1)(x - 3)(x^4 - 2x^3 - 4x^2 + 8x - 6) - (x - 1)(x - 3)(x^2 - 1) + (x - 1)(x + 1)(x^2 - 2x - 3)] / [(x^2 - 1)(x^4 - 2x^3 - 4x^2 + 8x - 6)]
Expanding the numerator and collecting like terms, we get:
Perimeter = [x^6 - 6x^5 - 19x^4 + 34x^3 + 30x^2 - 38x + 6] / [(x^2 - 1)(x^4 - 2x^3 - 4x^2 + 8x - 6)]
Thus, the simplified expression for the perimeter of the triangle in terms of x is:
Perimeter = (x^6 - 6x^5 - 19x^4 + 34x^3 + 30x^2 - 38x + 6) / [(x^2 - 1)(x^4 - 2x^3 - 4x^2 + 8x - 6)]
To find the perimeter of the triangle, you need to add up the lengths of all three sides. Let's start by simplifying the given expressions for each side of the triangle.
Side 1: 1/(x - 1)
Side 2: -1/(x^2 - 1)
Side 3: 1/(x^2 - 2x - 3)
To add these fractions, we need a common denominator. The first step is to factorize the denominators of each side:
(x - 1) can be factored further as it is already in simple form.
(x^2 - 1) is a difference of squares, so it can be factored as (x - 1)(x + 1).
(x^2 - 2x - 3) can be factored as (x - 3)(x + 1).
Let's now consider the common denominator for all three sides. We need to take the product of all the factors in each denominator, excluding any duplicate factors:
Common denominator = (x - 1)(x - 3)
Now, we can rewrite each of the fractions with the common denominator:
Side 1: (1/(x - 1)) * ((x - 3)/(x - 3))
= (x - 3) / ((x - 1)(x - 3))
= (x - 3) / (x^2 - 4x + 3)
Side 2: (-1/(x^2 - 1)) * ((x - 1)/(x - 1))
= (-1(x - 1))/ ((x^2 - 1)(x - 1))
= -(x - 1) / (x^3 - x - x + 1)
= -(x - 1) / (x^3 - 2x - 1)
Side 3: (1/(x^2 - 2x - 3)) * ((x - 1)/(x - 1))
= (x - 1) / ((x^2 - 2x - 3)(x - 1))
= (x - 1) / (x^3 - 3x^2 + x^2 + 3x - 3x - 3)
= (x - 1) / (x^3 - 2x^2 - 6)
Now that we have the simplified expressions for each side, we can calculate the perimeter by adding all three sides:
Perimeter = Side 1 + Side 2 + Side 3
= (x - 3) / (x^2 - 4x + 3) - (x - 1) / (x^3 - 2x - 1) + (x - 1) / (x^3 - 2x^2 - 6)
This is the simplified expression for the perimeter of the triangle in terms of x.