Question 122543
{{{(7x-8)/(x^2-2x)}}} Start with the given expression.



{{{(7x-8)/(x(x-2))}}} Factor the denominator.



{{{(7x-8)/(x(x-2))=A/(x)+B/(x-2)}}} Break up the fraction.



{{{cross((x(x-2)))((7x-8)/cross((x(x-2))))=(x(x-2))(A/cross(x)+B/cross(x-2))}}} Multiply both sides by the LCD {{{x(x-2)}}}. Doing this will eliminate every fraction.



{{{7x-8=A(x-2)+B(x)}}} Distribute and simplify.




Now let {{{x=0}}}


{{{7(0)-8=A(0-2)+B(0)}}}  Plug in {{{x=0}}}



{{{-8=-2A}}}  Simplify. Notice the B term goes away



{{{4=A}}}  Divide both sides by -2 to solve for A



So one numerator is {{{A=4}}} which means the first fraction is {{{4/x}}}



However since the fraction {{{4/x}}} is not in the list, this means the second fraction must be in the list





Now let {{{x=2}}}


{{{7(2)-8=A(2-2)+B(2)}}}  Plug in {{{x=0}}}



{{{6=2B}}}  Simplify. Notice the A term goes away



{{{3=B}}}  Divide both sides by 2 to solve for B



So another numerator is {{{B=3}}} which means the second fraction is {{{3/(x-2)}}}



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Answer:


So this shows us that the answer is {{{3/(x-2)}}} which means the choice is E)