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Add rational expressions with unlike denominators

Adding rational expressions with unlike denominators means rewriting each expression with an equivalent common least common denominator, typically found by factoring polynomial denominators and using each factor with its greatest required multiplicity. The learner combines numerators only after establishing a common denominator, simplifies the result, and preserves restrictions excluding values that make an original denominator zero; the scope includes numerical and polynomial expressions with linear or common quadratic factors, not abstract rational-function fields or partial-fraction decomposition.

Detailed Explanation: Add rational expressions with unlike denominators

To add rational expressions with unlike denominators:

  1. Factor each denominator completely.
  2. State the restrictions—values that make an original denominator equal to zero.
  3. Find the least common denominator (LCD) by using every factor with the greatest multiplicity.
  4. Rewrite each expression with the LCD.
  5. Add the numerators, keeping the common denominator.
  6. Simplify and keep the original restrictions.

Consider the example

3x2−9+2x+3.\frac{3}{x^2-9}+\frac{2}{x+3}.

Step 1: Factor the denominators.

The first denominator is a difference of squares:

x2−9=(x−3)(x+3).x^2-9=(x-3)(x+3).

So the expression becomes

3(x−3)(x+3)+2x+3.\frac{3}{(x-3)(x+3)}+\frac{2}{x+3}.

Step 2: State the restrictions.

The original denominators cannot equal zero:

x−3≠0andx+3≠0.x-3\ne 0 \quad\text{and}\quad x+3\ne 0.

Therefore,

x≠3,x≠−3.x\ne 3,\qquad x\ne -3.

Step 3: Find the LCD.

The LCD is

(x−3)(x+3).(x-3)(x+3).

The first fraction already has this denominator. The second fraction is missing the factor x−3x-3.

Step 4: Rewrite both fractions using the LCD.

Multiply the second fraction by x−3x−3\frac{x-3}{x-3}:

3(x−3)(x+3)+2(x−3)(x+3)(x−3).\frac{3}{(x-3)(x+3)} + \frac{2(x-3)}{(x+3)(x-3)}.

Now both fractions have the same denominator.

Step 5: Add the numerators.

3+2(x−3)(x−3)(x+3).\frac{3+2(x-3)}{(x-3)(x+3)}.

Simplify the numerator:

3+2(x−3)=3+2x−6=2x−3.3+2(x-3)=3+2x-6=2x-3.

Therefore,

2x−3(x−3)(x+3),x≠3,−3.\boxed{\frac{2x-3}{(x-3)(x+3)}}, \qquad x\ne 3,-3.

Only the numerators are added; the common denominator stays the same. The restrictions come from the original expression and must be kept even after simplifying.

Learn by doing: Add rational expressions with unlike denominators

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    Rational Expressions - Add/Subtract Unlike Denominators (Binomials), Final Answer

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