Daniel B. answered 12/03/22
A retired computer professional to teach math, physics
"Rational equation" means that y = P(x)/Q(x) for some polynomials P and Q.
"Vertical asymptote at x=5" means Q(5) = 0. (1)
"Vertical asymptote at x=-2" means Q(-2) = 0. (2)
"x-intercept at x=-1" means P(-1) = 0 (3)
"x-intercept at x=-5" means P(-5) = 0 (4)
"y-intercept at 4" means P(0)/Q(0) = 4 (5)
The polynomial of least order satisfying (1) and (2) is of the form
Q(x) = a(x-5)(x+2) for some constant a.
The polynomial of least order satisfying (3) and (4) is of the form
P(x) = b(x+1)(x+5) for some constant b.
For P(x)/Q(x) to satisfy (5) means
b(0+1)(0+5)/a(0-5)(0+2) = 4
Simplify into
5b/(-10a) = 4
b/a = -8 (6)
Any non-zero values for a and b satisfying (6) yield a solution.
An example of a solution is
y = -8(x+1)(x+5)/(x-5)(x+2)
Maria M.
Thank you so much, I greatly appreciate it.12/06/22