Question 121738


{{{(8x-1)/((x-2)(x+1))}}} Start with the given expression.



{{{(8x-1)/((x-2)(x+1))=A/(x-2)+B/(x+1)}}} Break up the fraction.



{{{cross(((x-2)(x+1)))((8x-1)/cross(((x-2)(x+1))))=((x-2)(x+1))(A/(x-2)+B/(x+1))}}} Multiply both sides by the LCD {{{(x-2)(x+1)}}}. Doing this will eliminate every fraction.



{{{8x-1=A(x+1)+B(x-2)}}} Distribute and simplify.


Now let {{{x=2}}}


{{{8(2)-1=A(2+1)+B(2-2)}}} Plug in {{{x=2}}}


     

{{{15=A(3)+B(0)}}} Simplify. Notice how the B term is multiplied by zero. So this cancels out B.



{{{15=3A}}} Multiply



{{{15/3=A}}} Divide both sides by 3 to isolate A




So our first numerator is {{{A=5}}}



So the first fraction is {{{5/(x-2)}}} 



However, notice how {{{5/(x-2)}}} is <b>not</b> in the list of answers. So the answer must be the second fraction. So let's solve for B


     


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     Now let {{{x=-1}}}


{{{8(-1)-1=A(-1+1)+B(-1-2)}}} Plug in {{{x=-1}}}


     


{{{-9=A(0)+B(-3)}}} Simplify. Notice how the A term is multiplied by zero. So this cancels out A.




{{{-9=-3B}}} Multiply 



{{{-9/-3=B}}} Divide both sides by -3 to isolate B




So our second numerator is {{{B=3}}}




So the second fraction is {{{3/(x+1)}}} 
     





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Answer:



So one of the fractions that {{{(8x-1)/((x-2)(x+1))}}} decomposes into is {{{3/(x+1)}}} 




So the answer is D)