Reducing Algebraic Fractions
Reducing Algebraic Fractions is an intermediate algebra skill in Yoshiwara Intermediate Algebra, Chapter 8: Polynomial and Rational Functions. Students simplify rational expressions by factoring the numerator and denominator and canceling common factors.
Key Concepts
Property Fundamental Principle of Fractions: We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one. $$\frac{a \cdot c}{b \cdot c} = \frac{a}{b} \quad \text{if} \quad b, c \neq 0$$ To reduce an algebraic fraction: 1. Factor the numerator and the denominator. 2. Divide the numerator and denominator by any common factors. Caution: We can cancel common factors , but we cannot cancel common terms .
Examples To reduce $\frac{12x^5y^2}{8x^3y^3}$, we find the common factor $4x^3y^2$. Factoring gives $\frac{3x^2 \cdot 4x^3y^2}{2y \cdot 4x^3y^2}$, which simplifies to $\frac{3x^2}{2y}$. The fraction $\frac{x+4}{x+8}$ cannot be reduced. The $x$ is a term, not a factor, so it cannot be canceled. To reduce $\frac{7x+14}{21}$, first factor the numerator and denominator: $\frac{7(x+2)}{7(3)}$. Canceling the common factor of 7 leaves $\frac{x+2}{3}$.
Explanation To simplify an algebraic fraction, you must first factor the top and bottom completely. Then, you can cancel out identical factors. Remember, you can only cancel parts that are multiplied, not parts that are added or subtracted.
Common Questions
How do you reduce an algebraic fraction?
Factor both the numerator and denominator completely, then cancel any common factors. The result is the simplified form of the rational expression.
What is an example of reducing an algebraic fraction?
Reduce (6x squared plus 12x) divided by (3x). Factor: 6x(x plus 2) divided by 3x. Cancel 3x: result is 2(x plus 2).
What conditions must be noted when reducing algebraic fractions?
State any restrictions on the variable: values that make the original denominator zero are excluded. For example, x cannot equal 0 in the example above.
Why must you factor before canceling?
You can only cancel common factors, not terms. For example, in (x plus 3) divided by (x plus 5), you cannot cancel x because it is not a factor of each expression.
What chapter covers reducing algebraic fractions in Yoshiwara Intermediate Algebra?
Reducing algebraic fractions is covered in Chapter 8: Polynomial and Rational Functions in Yoshiwara Intermediate Algebra.