Find constants A and B such that
x−7x2−x−2=Ax−2+Bx+1
for all x such that $x \neq -1$ and $x \neq 2$. Give your answer as the ordered pair (A,B).
+309 Answer: (−53,83)
Solution:
The right side of the equation can be turned into A(x+1)+B(x−2)x2−x−2. After that, you can multiply both sides of the equation to get x−7=A(x+1)+B(x−2).
The coefficient of the x on the left, which is one, is equal to A + B, because the x term on the left side is made up of Ax + Bx. The constant term on the left side, -7, is equal to A-2B. This is because when expanding out the terms on the right, you would get Ax +A+Bx −2B. From this information, you get two equations:
A+B=1
and
A−2B=−7
Subtracting the second equation from the first gives 3B = 8, which means that B = 8/3. Substituting this value into the first equation gives A + 8/3 = 1. Solving this gives A = 1-8/3 = -5/3.
Putting this into an ordered pair gives (−53,83).
