Algebra

Find a rational function that satisfies the given conditions.

Vertical asymptotes x=-2,x=7

Horizontal asymptote y=7/2

​x-intercept ​(−5​, ​0)

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  1. vertical asymptotes: 1/((x+2)(x-7))
    root at x = -5:

    (x+5)/((x+2)(x-7))

    horizontal asymptote. we need the degree to be equal, so try

    7x(x+5)
    --------------
    2(x+2)(x-7)

    The problem here is that now we have another x-intercept at (0,0)

    So, let's work with

    7(x+5)^2
    --------------
    2(x+2)(x-7)

    That gives us an intercept, but it does not cross the x-axis.

    We could go with something like

    7(x^2+1)(x+5)
    --------------
    2(x+2)(x-7)^2

    That gives us the same degree top and bottom, a single crossing at -5, and still the two vertical asymptotes.

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