Operations with Rationals

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More Lessons for Intermediate Algebra or Algebra II

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

• the definition and domain of a rational expression
• how to multiply and divide rational expressions
• how to add and subtract rational expressions
• how to simplify complex fractions

Definition and Domain of a Rational Expression

We have rational expressions whenever we have a fraction that has a polynomial in the numerator and/or in the denominator. An excluded value in the function is any value of the variable that would make the denominator equal to zero. To find the domain, list all the values of the variable that, when substituted, would result in a zero in the denominator.
How to define a rational expression. Defining Rational Expressions

Multiplying and Dividing Rational Expressions

Dividing rational expressions is basically two simplifying problems put together. When dividing rationals, we factor both numerators and denominators and identify equivalents of one to cancel. After identifying these equivalents, we take the reciprocal of the second fraction and divide. Multiplying rational expressions is the same as dividing rationals, except that we do not take the reciprocal of the second fraction.
How to multiply and divide rational expressions. This video explains how to multiply and divide rational expressions.

Rational Expressions: Multiplying and Dividing. Ex 1.
This video shows how to divide two rational expressions containing binomials. Rational Expressions: Multiplying and Dividing. Ex 2.
This video shows how to divide two rational expressions containing quadratic trinomials.

Rational Expressions: Multiplying and Dividing. Ex 3.
This videos shows how to divide two rational expressions

Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions is similar to adding fractions. When adding and subtracting rational expressions, we find a common denominator and then add the numerators. To find a common denominator, factor each first. This strategy is especially important when the denominators are trinomials.
This video explains how to add and subtract rational expression with and without like denominators. This video provides two examples of how to add and subtraction rational expressions with the the same denominators. This video provides two examples of how to add and subtraction rational expressions when the denominators are different.

This video provides two examples of how to add and subtraction rational expressions when the denominators are different.

Simplifying Complex Fractions

Simplifying rational expressions combines everything learned about factoring common factors and polynomials. When simplifying complex fractions, factor the numerator and denominator into terms multiplying each other and look for equivalents of one (something divided by itself). Include parenthesis around any expression with a "+" or "-" and if all terms cancel in the numerator, there is still a one there.
How to simplify a complex fraction. Simplifying Complex Fractions - Ex 1
This video shows how to simplify a complex fraction by getting common denominators, flipping, multiplying, and canceling. Simplifying Complex Fractions - Ex 2
This video shows how to simplify a complex fraction by getting common denominators, flipping, multiplying, and canceling. Simplifying Complex Fractions - Ex 3
This video shows how to simplify a complex fraction by getting common denominators, flipping, multiplying, and canceling.