Rational Expressions & Simplification

Calculus Readiness · Axiom Academy

A rational expression is just a fraction of polynomials — and it follows the very same rules: factor, cancel common factors, and combine over a common denominator. 1. Simplifying — Factor and Cancel To simplify, factor the top and bottom, then cancel any factor that appears in both. A factor cancels because — so a matched factor on top and bottom simply leaves a 1 behind. Factor top and bottom, then cancel the shared (x+3) To multiply , factor everything first and cancel any factor on a top against a matching factor on a bottom — anywhere in the product — before you multiply across. To divide , keep–change–flip: leave the first fraction, turn into , and flip the second. . Cancel shared factors across the whole product first. . Flip the divisor, then multiply. A factor on any numerator cancels a matching factor on any denominator. Exclude every x that zeroed an original denominator — and the divisor's numerator. . Flip and multiply to get ; the (x+2) and (x-1) cancel, leaving x+1 . 3. Adding & Subtracting — Common Denominator Just like numerical fractions, you cannot add until the denominators match. Build the least common denominator (LCD) — the product of all distinct factors — rescale each fraction to it, then add the numerators. Find the LCD: with denominators x and x+1 (no shared factor), the LCD is their product x(x+1) . Rescale, then add: multiply each fraction by the factor it lacks, so both ride x(x+1) , then combine the numerators into a single fraction.

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