The fraction (5x-11)/(2x^2+x-6) was obtained by adding the two fractions Ax+2 and B2x-3. The values of A and B must be, respectively:

1956 AMC 12 Problem 17

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17.

The fraction

\dfrac{5x-11}{2x^2+x-6}

was obtained by adding the two fractions

\dfrac A{x+2}

and

\dfrac B{2x-3}.

The values of

A

and

B

must be, respectively:

5x,

-11

-11,

5x

-1,

3

3,

-1

5,

-11

Factor

2x^2+x-6=(x+2)(2x-3)

After combining the two fractions, match coefficients in

A(2x-3)+B(x+2)=5x-11

Combining the proposed partial fractions gives

\frac{A(2x-3)+B(x+2)}{(x+2)(2x-3)}.

Matching its numerator with

5x-11

yields

\begin{aligned} 2A+B&=5,\\ -3A+2B&=-11. \end{aligned}

Solving gives

A=3

and

B=-1.

Thus, the correct answer is D.