## Rational Expressions

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1. Definition

2. Evaluating Rational Expressions

3. Domain of Rational Expressions

4. Simplify Rational Expressions

5. Multiply Rational Expressions

6. Divide Rational Expressions

7. Add/Subtract Rational Expressions with SAME Denominators

8. Add/Subtract Rational Expressions with DIFFERENT Denominators

9. Solve Rational Equations

10. Complex Fractions

### 1. Definition

A rational expression is a quotient of two polynomials.

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### 2. Evaluating Rational Expressions

To evaluate a rational expression, we replace the variables with given values.

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### 3. Domain of Rational Expressions

Domain of an expression is the set of values that the variables can be replaced with.

The domain of a rational expression is the set of all real numbers except the values that make the denominator equal to zero.

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### 4. Simplify Rational Expressions

1. Factor the numerator and the denominator, if they are not factored.

2. Divide out common factors.

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### 5. Multiply Rational Expressions

1. Factor the numerators and the denominators, if they are not factored.

2. Divide out common factors.

3. Multiply the remaining factors in the numerators.

4. Multiply the remaining factors in the denominators.

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### 6. Divide Rational Expressions

Divide an expression is to multiply its reciprocal.

To divide rational expressions:

1. Change the division sign to multiplication sign.

2. Change the divisor to its reciprocal.

3. Multiply the expressions, as shown above.

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### 7. Add/Subtract Rational Expressions with SAME Denominators

To add/subtract rational expressions with common denominators:

1. Add/Subtract the numerators.

2. Rewrite the common denominator.

3. Simplify, if possible.

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### 8. Add/Subtract Rational Expressions with DIFFERENT Denominators

To add/subtract rational expressions with different denominators:

1. Find the Least Common Multiple (LCM) of the denominators( this might involve factoring )

The coefficient is the LCM of all coefficients

Each variable factor is raised to the highest power among them.

2. Multiply the numerator and the denominator of each rational term with a proper factor (LCM/denominator)

The new rational expressions are now have common denominators.

3. Add/Subtract the new expressions.

4. Simplify, if possible.

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### 9. Solve Rational Equations

1. Find the Least Common Multiple (LCM) of the denominators(as shown above).

2. Multiply each rational term (the numerator only) with the LCM.

3. Simplify each rational term (divide out all denominators).

4. Solve the new equation.

5. Check the solutions against the domains of all rational terms.

**Special Case: Solving Proportions**

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### 10. Complex Fractions

Complex fractions are rational expressions whose numerator or denominator (or both) are rational expressions.

To simplify a complex fraction

1. Find the common denominator of all terms.

2. Multiply each term (the numerator only) with the common denominator.

3. Simplify each term (divide out all denominators).

4. Simplify the numerator and denominator (remove grouping symbols, combine like terms).