Algebra 2 - Unit 7: Rational Expression

Solving real-world problems

When a real-world situation can be modeled by a rational function, the answer to a real-world problem usually comes from solving the rational equation.

Be careful when subtracting!

When subtracting, remember to use parentheses and to distribute the subtraction across the entire numerator.

Holes in the Graph

A hole in the graph, represented by an open circle, indicates a discontinuity at a single point. Such discontinuities can usually be removed by reducing the rational expression.

How are rational expressions like and unlike rational numbers?

A rational expression is like a rational number, in that it contains a numerator divided by a denominator. Unlike a rational number, the numerator and denominator of a rational expression can contain polynomials as well as numbers.

LESSON 7.3:

Adding and Subtracting Rational Expressions

LESSON 7.8:

Applications of Rational Equations

LESSON 7.6:

Asymptotes of Rational Function

Discontinuities on the Graph of a Rational Function

At a value that is not in the domain of a rational function, we have one of two types of discontinuities: either a vertical asymptote (which we will learn more about in the next lesson) or a hole.

What does the denominator tell you?

Because division by zero is undefined, the only numbers that are not in the domain of a rational function are the zeros of the denominator.

LESSON 7.5:

Discontinuities of Rational Functions

Many real-world situations

For example, involving mixtures, work, and comparing rates — can be modeled using rational functions.

Adding rational expressions with different denominators

If two or more rational expressions have different denominators, first rewrite them all as equivalent fractions with a common denominator.

Adding rational expressions with the same denominator

If two or more rational expressions have the same denominator, to add (or subtract) them, just add (or subtract) their numerators.

Extraneous Solutions

In the process of solving rational equations, we sometimes encounter extraneous solutions — that is, solutions that do not actually work in the original problem because they make one or more fraction undefined. Always look back to the original problem to identify and rule out extraneous solutions!

Canceling before multiplying

It almost always makes your work easier if you factor the numerators and denominators of both fractions and cancel common factors before you multiply, rather than trying to factor and reduce after you multiply.

Factor First

It is often helpful to first factor all numerators and denominators before cross-multiplying, and to keep track of common factors rather than multiply the entire expression out.

Rational expressions are like rational numbers

Just like rational numbers, rational expressions can be added, subtracted, multiplied and divided (although you can't divide by zero).

LESSON 7.2:

Multiplying and Dividing Rational Expressions

Solving Rational Equations

Rational equations can be solved by using cross-multiplication to rewrite them as polynomial equations.

Multiplying rational expressions

Rational expressions are multiplied the same way as any other fractions: multiply the numerators and multiply the denominators. Steps to Multiply Rational Fractions: 1. Factor each numerator and denominator completely. 2. Simplify by dividing out common factors. 3. Multiply the numerators and denominators.

LESSON 7.4:

Simplifying Complex Fractions

LESSON 7.1:

Simplifying Rational Expressions

LESSON 7.7:

Solving Rational Functions

Interpreting Solutions

Some solutions found algebraically won't be meaningful in the context of the problem. Always look back and think about what your answers mean!

Domain of a rational function

The domain of a rational function is the set of all values of the input variable that can be used to compute the function.

End Behavior

The end behavior of a rational function depends on the degrees of the numerator and the denominator: If the degree of the denominator is higher than the degree of the numerator, in the long run, the function will trend towards zero, so there is a horizontal asymptote at y = 0. If the degree of the numerator and denominator are equal, then in the long run the function will trend toward a constant — in particular, towards the ratio of the leading coefficients. So there will be a horizontal asymptote of the form y = c, where c is that ratio. If the degree of the numerator is greater than the degree of the denominator by exactly one, then in the long run the function will trend towards a straight line. We call this a slant asymptote. (This is also called an oblique asymptote.) If the degree of the numerator exceeds the degree of the denominator by more than one, then in the long run the function will trend towards a parabola, cubic, or higher-degree monomial.

Method 1

The first method is to rewrite the numerator as a single rational expression, then rewrite the denominator as a single rational expression, and finally divide.

Method 2

The second method is to multiply everything by the least common multiple of all of the denominators of the "inner" fractions.

Vertical Asymptotes

The vertical asymptotes of a rational function correspond to the zeros of its denominator (after all removable discontinuities have been accounted for).

X-intercepts

The x-intercepts of a rational function correspond to the zeros of its numerator (after all removable discontinuities have been accounted for).

Y-intercepts

The y-intercepts of a rational function can be found by setting x = 0.

Dividing rational expressions

To divide rational expressions, just invert the second one and multiply. Steps to Divide Rational Fractions: 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second. 2. Factor the numerators and denominators completely. 3. Multiply the numerators and denominators together. 4. Simplify by dividing out common factors.

Finding a common denominator

To find a common denominator, factor both denominators and see which factors are present in some rational expressions but not in others.

How do we reduce a rational expression to lowest terms?

To reduce a rational expression to lowest terms, first completely factor the numerator and denominator. Then cancel out any factors that appear in both the numerator and denominator.

Equivalent Rational Expressions

Two equivalent rational expressions will agree with each other at all values of the variable for which both are defined, but there may be points where one rational expression is defined and the other is not.

When are two rational expressions equivalent to one another?

Two rational expressions are equivalent to one another if their cross-products are equal. Another way to check whether two expressions are equivalent is to reduce them both to the lowest terms.

Simplifying complex rational expressions

You can simplify complex rational expressions (also called "nested rational expressions") using two different methods.