Add or subtract as indicated. Express your result in simplest form.
(4y)/(y^2+6y+5) plus (2y)/(y^2-1)
My answer is: (6y)/((y+5)(y-1))
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Geometry. Refer to the triangle in the figure. Find an expression that represents its perimeter
It has the following: (3)/(4x), (5)/(x^2),(1)/(x^2)
My answer: by using the formula for permiter that is P=a+b+c
My answer for this is: (3(8+x))/(4x^2)
first: I don't agree with the denominator: shouldn't it be
(y+1 )*(y-1)(y+5)
second correct.
For the first question, to add or subtract fractions, you need to have a common denominator. In this case, the common denominator is (y^2 + 6y + 5)(y^2 - 1).
To express (4y)/(y^2 + 6y + 5) and (2y)/(y^2 - 1) with a common denominator, you need to multiply the numerator and denominator of each fraction by the missing factors in their respective denominators.
For the first fraction, the missing factor is (y^2 - 1):
(4y)/(y^2 + 6y + 5) = (4y/(y^2 + 6y + 5)) * ((y^2 - 1)/(y^2 - 1))
= (4y(y^2 - 1))/((y^2 + 6y + 5)(y^2 - 1))
For the second fraction, the missing factor is (y^2 + 6y + 5):
(2y)/(y^2 - 1) = (2y/(y^2 - 1)) * ((y^2 + 6y + 5)/(y^2 + 6y + 5))
= (2y(y^2 + 6y + 5))/((y^2 - 1)(y^2 + 6y + 5))
Now, we can add the two fractions together:
(4y(y^2 - 1))/((y^2 + 6y + 5)(y^2 - 1)) + (2y(y^2 + 6y + 5))/((y^2 - 1)(y^2 + 6y + 5))
Combining the terms in the numerator:
= (4y(y^2 - 1) + 2y(y^2 + 6y + 5))/((y^2 + 6y + 5)(y^2 - 1))
= (4y^3 - 4y + 2y^3 + 12y^2 + 10y)/((y^2 + 6y + 5)(y^2 - 1))
= (6y^3 + 12y^2 + 6y)/((y^2 + 6y + 5)(y^2 - 1))
Now, simplifying the expression, we can factor out a common factor of 6y:
= (6y(y^2 + 2y + 1))/((y^2 + 6y + 5)(y^2 - 1))
= (6y(y + 1)^2)/((y + 5)(y - 1)(y + 1))
So, the simplified form of the expression is (6y)/((y + 5)(y - 1)).
For the second question, to find the perimeter of a triangle, you need to add the lengths of all three sides.
The given lengths are (3)/(4x), (5)/(x^2), and (1)/(x^2).
Using the formula for the perimeter of a triangle P = a + b + c, where a, b, and c represent the lengths of the sides, we can substitute the given lengths:
P = (3)/(4x) + (5)/(x^2) + (1)/(x^2)
To simplify this expression, we need to find a common denominator. The common denominator for (4x), (x^2), and (x^2) is (4x^2).
Multiplying each fraction by the appropriate factor to get the common denominator:
P = (3(1))/(4x(1)) + (5(4))/(x^2(4)) + (1(4))/(x^2(4))
= (3)/(4x) + (20)/(4x^2) + (4)/(4x^2)
Now, combining like terms in the numerator:
P = (3 + 20 + 4)/(4x^2)
Simplifying:
P = (27)/(4x^2)
So, the expression that represents the perimeter of the triangle is (27)/(4x^2).