Simplifying algebraic fractions means rewriting a rational expression in an equivalent form with no common factors left in the numerator and denominator. This skill matters because it makes expressions easier to evaluate, graph, compare, and use in equations. The main idea is the same as reducing ordinary fractions, but variables bring extra conditions that must be tracked carefully.
Key Facts
- An algebraic fraction is simplified by canceling common factors, not common terms.
- Factor first: x^2 - 9 = (x - 3)(x + 3).
- If A, B, and C are expressions and C ≠ 0, then AC/BC = A/B.
- Excluded values come from the original denominator before canceling.
- (x^2 - 5x + 6)/(x^2 - 4) = ((x - 2)(x - 3))/((x - 2)(x + 2)) = (x - 3)/(x + 2), with x ≠ 2 and x ≠ -2.
- A simplified expression is equivalent to the original only for values that are allowed in the original expression.
Vocabulary
- Algebraic fraction
- A fraction whose numerator, denominator, or both contain variables.
- Rational expression
- An algebraic expression that can be written as a quotient of two polynomials.
- Factor
- A quantity that is multiplied by another quantity to make a product.
- Common factor
- A factor that appears in both the numerator and the denominator of a fraction.
- Excluded value
- A value of the variable that would make the original denominator equal to zero.
Common Mistakes to Avoid
- Canceling terms instead of factors is wrong because cancellation only applies to multiplied factors. In (x + 2)/(x + 5), the x terms cannot be canceled.
- Forgetting to factor completely is wrong because hidden common factors may remain. For example, x^2 - 4 must be factored as (x - 2)(x + 2) before simplifying.
- Ignoring excluded values is wrong because canceled factors can still represent values that made the original expression undefined. In (x - 3)/(x^2 - 9), x = 3 and x = -3 must both be excluded before simplifying.
- Canceling a factor equal to zero without restrictions is wrong because division by zero is undefined. When canceling (x - 4), the condition x ≠ 4 must be kept.
Practice Questions
- 1 Simplify (6x^2y)/(9xy^3) and state any restrictions on x and y.
- 2 Simplify (x^2 + 7x + 12)/(x^2 - 16) and state all excluded values.
- 3 Explain why (x + 3)/(x + 5) cannot be simplified by canceling x, but (x(x + 3))/(x(x + 5)) can be simplified with a restriction.