To simplify fractions with variables (called rational expressions), follow these steps—using the image’s examples and key principles:

Core Idea

Simplifying variable fractions is like simplifying regular fractions: cancel common factors (not terms!) between the numerator and denominator. But first, you need to factor both parts completely.

Step-by-Step Guide

1. Factor the Numerator & Denominator

Break down both into their simplest factors:

• Factor out common constants/variables: e.g., (2x +4 =2(x+2)), (2x+6=2(x+3)).
• Use special rules: Like difference of squares ((x^2 -a^2=(x-a)(x+a))) or trinomial factoring ((x^2+bx+c=(x+m)(x+n))).

2. Cancel Common Factors

If a factor appears in both the numerator and denominator, cancel it (since (\frac{f}{f}=1) for non-zero (f)).

3. Note Domain Restrictions

Any value of the variable that makes the original denominator zero is undefined (even if the factor cancels out).

Example from the Image

Let’s simplify (\frac{2x+4}{2x+6}):

1. Factor:
   • Numerator: (2(x+2))
   • Denominator: (2(x+3))
2. Cancel: The common factor is (2), so we get (\frac{x+2}{x+3}).
3. Restriction: (x≠-3) (original denominator is zero when (x=-3)).

Another Example

Simplify (\frac{x^2 -4}{x^2 +x -6}):

1. Factor:
   • Numerator: (x^2-4=(x-2)(x+2)) (difference of squares).
   • Denominator: (x^2+x-6=(x+3)(x-2)) (trinomial factor).
2. Cancel: Common factor ((x-2)), so (\frac{x+2}{x+3}).
3. Restrictions: (x≠-3) and (x≠2) (original denominator zero at these values).

Key Mistake to Avoid

Don’t cancel terms directly! For example, you can’t cancel (x) in (\frac{x+2}{x+3})—(x) is a term inside parentheses, not a factor of the entire numerator/denominator.

This method works for all rational expressions—just remember to factor first, cancel factors, and note restrictions!

作者声明:本文包含人工智能生成内容。