Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
Divide rational expressions with denominator and zero-divisor restrictions
Problem
Divide and fully simplify \([(x^2~-~9)/(x~+~1)]\) by \([(x~-~3)/(x~+~1)]\). State the restrictions from the original denominators and from the divisor being nonzero.
Big Picture
What this problem is really about
Dividing rational expressions creates two kinds of restrictions before any reciprocal is taken. Exclude zeros of all original denominators, and also exclude inputs that make the entire divisor rational expression equal zero. Then multiply by the reciprocal, factor, and cancel on that allowed domain; every original exclusion remains even if its factor disappears.
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