1. ## 1 math question

Identify the form of the partial fraction decomposition. Do not solve for the constants. 8x2+7x
(x+5)3

A: A + Bx + Cx2
X+5 (x+5)2 (x+5)3

B: A + B + C
X+5 (x+5)2 (x+5)3

C: Ax2 + Bx + C
(x+5)3

D: A + Bx+C
X+5 (x+5)2

Identify the conic section with the given equation.
5x2-6y2-9x+2y+3=0

p.s. all the two and threes after the par. and x's are supposed to be raised to that power!

2. Hello, Lane!

Identify the form of the partial fraction decomposition for: .$\displaystyle \frac{8x^2+7x}{(x+5)^3}$

$\displaystyle A\!:\;\frac{A}{x+5} + \frac{Bx}{(x+5)^2} + \frac{Cx^2}{(x+5)^3}$ . . . $\displaystyle B\!:\;\;\frac{A}{x+5} + \frac{B}{(x+5)^2} + \frac{C}{(x+5)^3}$

. . . $\displaystyle C\!:\;\;\frac{Ax^2 + Bx + C}{(x+5)^2}$. . . . . $\displaystyle D\!:\;\;\frac{A}{x+5} + \frac{Bx+C}{(x+5)^2}$

With repeated linear factors,
. . we need a linear fraction for "each power".

Since $\displaystyle x+5$ is cubed, the denominators must be: .$\displaystyle x+5,\;(x+5)^2,\;(x+5)^3$

The answer is choice $\displaystyle B.$

3. Hello, Lane!

We can "eyeball" the second one . . .

Identify the conic section: .$\displaystyle 5x^2-6y^2-9x+2y+3\:=\:0$

Since the $\displaystyle x^2$-term and the $\displaystyle y^2$-term have opposite signs,

. . the conic is a $\displaystyle \text{hyperbola.}$