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# What two rational expressions sum to 3x−2/x 2 −x−12 ?

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What two rational expressions sum to 3x−2/x2−x−12 ?

Jan 7, 2022

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To find which two rational expressions sum to  (3x - 2) / (x2 - x - 12):

1)  Factor the denominator:  (3x - 2) / [ (x - 4)(x + 3) ]

2)  This means that the two rational expressions are:  A / (x - 4)  and  B / (x + 3)

3)  Start with:  A / (x - 4)  +  B / (x + 3)

write as one fraction:  [ A(x + 3) + B(x - 4) ]  /  [ (x - 4)(x + 3) ]

4)  Simplify this:  =  ( Ax + 3A + Bx - 4B ) / [ (x - 4)(x + 3) ]

5)  When you compare this result with  (3x - 2) / [ (x - 4)(x + 3) ]

notice that the numerators must be equal:  ( Ax + 3A + Bx - 4B )  =  (3x - 2)

6)  Rearranging the numerator:  [ Ax + Bx + 3A - 4B ]  =  (3x - 2)

7)  This means that   Ax + Bx  =  3x   and   3A - 4B  =  -2

--->     Ax + Bx  =  3x     --->     A + B  =  3

--->                                        3A - 4B  =  -2

8)  Solving for A and B:  A = 10/7   and   B = 11/7

9)  Which means that the two rational expressions are:  ( 10/7 ) / (x - 4)   and   ( 11/7 ) / (x + 3)

Jan 8, 2022