Question

If $\frac{3x}{x^2+x-2} = \frac{A}{x+2} + \frac{B}{x-1}$, then the ordered pair $(A, B)$ is

• A. $(1, 2)$
• B. $(-1, 2)$
• C. $(2, -1)$
• D. $(2, 1)$

Hint:

For simple linear factors, use the "Cover-up Method." To find $A$, cover $(x+2)$ in the original fraction and plug $x = -2$ into the rest: $\frac{3(-2)}{-2-1} = \frac{-6}{-3} = 2$. It's much faster during exams!

Answer 1

The Correct Option is D

Step 1: Set up the Identity: The given equation is: $$\frac{3x}{(x+2)(x-1)} = \frac{A}{x+2} + \frac{B}{x-1}$$ Multiply both sides by the common denominator $(x+2)(x-1)$: $$3x = A(x-1) + B(x+2)$$

Step 2: Solve for A and B using the substitution method: To find $B$, let $x = 1$ (which eliminates the $A$ term): $$3(1) = A(1-1) + B(1+2)$$ $$3 = 3B \implies B = 1$$ To find $A$, let $x = -2$ (which eliminates the $B$ term): $$3(-2) = A(-2-1) + B(-2+2)$$ $$-6 = -3A \implies A = 2$$

Step 3: Form the Ordered Pair: The values found are $A = 2$ and $B = 1$. Therefore, the ordered pair $(A, B)$ is $(2, 1)$.