Question

Someone pls help me ASAP! Express the following as the sum of its partial fraction: 2x+4/x(x+2)(x-4)


`(2x + 4)/((x + 2)(x - 4))`
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Answer 1

`(2x+4)/(x(x+2)(x-4)) \equiv (1)/(2(x-4))-(1)/(2x)`

Explanation:

Partial Fractions

Write out the expression as an identity:

`\begin{aligned}(2x+4)/(x(x+2)(x-4)) & \equiv (A)/(x)+(B)/((x+2))+(C)/((x-4))\\\\\implies (x(2x+4)(x+2)(x-4))/(x(x+2)(x-4)) & \equiv (Ax(x+2)(x-4))/(x)+(Bx(x+2)(x-4))/((x+2))+(Cx(x+2)(x-4))/((x-4))\\\\\implies 2x+4 & \equiv A(x+2)(x+4)+ Bx(x-4)+Cx(x+2)\end{aligned}`

Calculate the values of A, B and C using substitution:

`\begin{aligned}2x+4 & = A(x+2)(x-4)+Bx(x-4)+Cx(x+2)\\\\x=4 \implies 12 & = A(0)+B(0)+C(24)\implies C=(1)/(2)\\\\x=-2 \implies 0 & = A(0)+B(12)+C(0) \implies B=0\\\\ x=0 \implies 4 & = A(-8)+B(0)+C(0) \implies A=-(1)/(2)\end{aligned}`

Replace A, B and C in the original identity:

`\begin{aligned}(2x+4)/(x(x+2)(x-4)) & \equiv (A)/(x)+ (B)/((x+2))+(C)/((x-4))\\\\& \equiv -(1)/(2x)+(1)/(2(x-4))\\\\\implies (2x+4)/(x(x+2)(x-4))& \equiv (1)/(2(x-4))-(1)/(2x)\end{aligned}`