Express the following as a sum of partial fraction
2x+4÷x^2(x-2)
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or, so you can do it on your own, you want to find
(2x+4)/(x^2(x-2)) = A/x + B/x^2 + C/(x-2)
so put the right side over the common denominator. That gives you
Ax(x-2) + B(x-2) + Cx^2 = 2x+4
Now expand and equate coefficients.
(A+C)x^2 + (-2A+B)x - 2B = 2x+4
A+C = 0
-2A+B = 2
-2B = 4
Solving that, you get
B = -2, A = -2, C = 2
(2x+4)/(x^2(x-2)) = -2/x - 2/x^2 + 2/(x-2)
To express the given expression, 2x + 4 / x^2(x - 2), as a sum of partial fractions, we need to break it down into simpler fractions. The denominator in our case can be factored as x^2(x - 2).
Let's consider the denominator first:
1. x^2: Since this is a quadratic term with a multiplicity of 2, the partial fraction form for this term will be:
Ax + B / x^2
2. (x - 2): This is a linear term, so the partial fraction form will be:
C / (x - 2)
Now, let's combine these two partial fractions into one:
2x + 4 / x^2(x - 2) = (Ax + B) / x^2 + C / (x - 2)
To find the values of A, B, and C, we'll start by getting a common denominator for the right side:
2x + 4 / x^2(x - 2) = [ (Ax + B)(x - 2) + Cx^2 ] / x^2(x - 2)
Next, we'll multiply both sides by the original denominator x^2(x - 2) to eliminate the denominators:
2x + 4 = (Ax + B)(x - 2) + Cx^2
Expanding the right side:
2x + 4 = Ax^2 - 2Ax + Bx - 2B + Cx^2
Combining like terms:
2x + 4 = (A + C)x^2 + (-2A + B)x + (-2B)
Now, let's equate the coefficients of the like terms on both sides:
For the x^2 term:
0x^2 on the left side = (A + C)x^2 on the right side
This leads to: A + C = 0 ...(1)
For the x term:
2x on the left side = (-2A + B)x on the right side
This leads to: -2A + B = 2 ...(2)
For the constant term:
4 on the left side = -2B on the right side
This leads to: -2B = 4 ...(3)
Now we have a system of three equations (1), (2), and (3) in variables A, B, and C.
Solving this system of equations, we find:
From equation (3), B = -2
From equation (2), A = -1
From equation (1), C = 1
Therefore, the partial fraction form for the given expression is:
2x + 4 / x^2(x - 2) = (-x - 2) / x^2 + 1 / (x - 2)
So, the expression can be expressed as a sum of partial fractions as follows:
2x + 4 / x^2(x - 2) = (-x - 2) / x^2 + 1 / (x - 2)