Question 1101554
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To have a pole at x=a, the denominator of the rational function must contain a factor of (x-a).<br>
In your example, with poles at x=1 and x=2, we need factors of (x-1) and (x-2) in the denominator.<br>
To get a zero at x=b, the numerator of the rational function must contain a factor of (x-b).<br>
In your example, with zeros at x=4 and x=5, we need factors of (x-4) and (x-5) in the numerator.<br>
To get a horizontal asymptote of y=3, we need the degrees of the numerator and denominator to be the same, and we need the leading coefficient in the numerator to be 3 times the leading coefficient in the denominator.<br>
The simplest rational function that satisfies all these requirements is<br>
{{{((3(x-4)(x-5))/(x-1)(x-2))}}}