# Manipulating of Rational Expressions

## Manipulating of Rational Expressions

# Expressing Rational Expressions

- A
**rational expression**is one that can be expressed as the fraction of two polynomials. - Working with rational expressions is similar to working with regular fractions.

# Simplifying Rational Expressions

- Always check if top and bottom of the fraction can be factored. If a same factor is found in both, it can be canceled out.
- Example:
`(3x^2 -12x)/3x`

simplifies as`(3x(x - 4))/3x`

and further simplifying gives`x - 4`

, after cancelling out`3x`

.

# Adding and Subtracting Rational Expressions

- To add or subtract rational expressions, they must have the same denominator, just like regular fractions.
- If two rational expressions do not have the same denominator, find a common denominator and transform them.
- Whenever you perform the operations, reflect the changes in the least common denominator.

# Multiplying and Dividing Rational Expressions

- When multiplying, simply multiply the numerators together and the denominators together. Aim to simplify afterwards.
- When dividing, remember to “flip” the second expression (reciprocal) and then multiply.

# Solving Equations with Rational Expressions

- Similar to equations with fractions, to clear the fraction, multiply by the common denominator.
- Be careful of values that make the denominator zero. Always cross-check with the original equation since these can be extraneous solutions.

# Graphing Rational Functions

- Sketching involves identifying key features like vertical asymptotes, horizontal asymptotes, and holes if any.
- Vertical asymptotes are found where the denominator equals zero, and horizontal asymptotes depend on the degree of the polynomials.

**The Golden Rule**: Always simplify a rational expression but never cancel terms.

Remember to **check your solutions** in the original question to ensure they don’t make any denominator equals zero, as they would be undefined.