1. Find constants $A$ and $B$ such that \[\frac{x + 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\] for all $x$ such that $x\neq -1$ and $x\neq 2$. Give your answer as the ordered pair $(A,B)$.
2. a) Suppose that \[|a - b| + |b - c| + |c - a| = 20.\] What is the maximum possible value of $|a - b|$?
b) Suppose that \[|a - b| + |b - c| + |c - d| + \dots + |m-n| + |n-o| + \cdots+ |x - y| + |y - z| + |z - a| = 20.\] What is the maximum possible value of $|a - n|$?
+128474 \( \frac{x + 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\)
We can use partial fraction decmposition to find A, B
Factoring the denominator on the left side we have (x - 2) ( x + 1)
Multiply through by this common denominator and we have
x + 7 = A(x + 1) + B(x - 2) simplify
x + 7 = Ax + A + BX - 2B equate coefficients and we get this system
A + B = 1
A - 2B = 7 subtract the second equation from the first
3B = -6 divide both sides by 3
B = -2
Using the first equation to find A
A + -2 = 1 add 2 to both sides
A = 3
So (A, B ) = ( 3, -2)
