Rewriting Rational Expressions
Aligned to A-APR.D.6 — Common Core State Standards for Mathematics.
What this lesson teaches
A rational expression can be rewritten as a quotient plus a remainder over the divisor, in the form q(x) + r(x)/b(x). Use polynomial long division, or factor and cancel when the numerator factors evenly.
Worked examples
Example 1
Rewrite (x^2 + 3x + 5)/(x + 1). Divide: x^2 + 3x + 5 by (x + 1). x times (x + 1) is x^2 + x, leaving 2x + 5; 2 times (x + 1) is 2x + 2, leaving remainder 3. So the result is x + 2 + 3/(x + 1).
Example 2
Simplify (x² − 9)/(x² + x − 6). Factor both: the top is (x − 3)(x + 3) and the bottom is (x + 3)(x − 2). Cancel the shared factor (x + 3) to get (x − 3)/(x − 2). Note the original was undefined at x = −3 and x = 2, and the simplified form must carry that restriction: x ≠ −3.
Example 3
Common mistake: cancelling terms instead of factors. In (x + 4)/x you cannot cross out the x to get 4 — cancelling is division, and it only applies to whole FACTORS multiplied together. Test with x = 4: (4 + 4)/4 = 2, not 4. Factor completely first; if a piece is joined to its neighbor by + or −, it cannot be cancelled.
Practice questions
- Simplify (x^2 − 4)/(x − 2) by factoring.
- Divide (x^2 + 5x + 6)/(x + 2).
- Rewrite (2x^2 + x − 1)/(x + 1) as quotient plus remainder.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Reescribir Expresiones Racionales
Una expresión racional se puede reescribir como un cociente más un residuo sobre el divisor, en la forma q(x) + r(x)/b(x). Usa la división larga de polinomios, o factoriza y cancela cuando el numerador se factoriza de forma exacta.
Ejemplo 1: Reescribe (x^2 + 3x + 5)/(x + 1). Divide x^2 + 3x + 5 entre (x + 1). x por (x + 1) es x^2 + x, deja 2x + 5; 2 por (x + 1) es 2x + 2, deja residuo 3. Así que el resultado es x + 2 + 3/(x + 1).
Ejemplo 2: Simplifica (x² − 9)/(x² + x − 6). Factoriza ambos: arriba queda (x − 3)(x + 3) y abajo (x + 3)(x − 2). Cancela el factor común (x + 3) para obtener (x − 3)/(x − 2). Observa que la expresión original no estaba definida en x = −3 ni en x = 2, y la forma simplificada debe conservar esa restricción: x ≠ −3.
Ejemplo 3: Error común: cancelar términos en vez de factores. En (x + 4)/x no puedes tachar la x para obtener 4 — cancelar es dividir, y solo se aplica a FACTORES completos multiplicados. Pruébalo con x = 4: (4 + 4)/4 = 2, no 4. Factoriza por completo primero; si una parte está unida a su vecina por + o −, no se puede cancelar.
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