Section 6.1 Rational Expressions
A rational expression is a ratio of two polynomials . In
other words, a rational expression
is of the form
, where
A and B are polynomials and B ≠ 0 .
If the polynomial B contains at least one variable term, the rational expression
is also
called an algebraic fraction.
Restricted Values
Recall that division by zero is undefined. Because rational expressions indicate
division
we must be careful to avoid denominators that are equal to zero. The values that
produce
a denominator of zero are called restricted values because they make the
rational
expression undefined.
To determine the restricted values of a rational function:
1. Set the denominator equal to zero.
2. Solve for the independent variable.
The values obtained are restricted from the domain.
Simplifying Rational Expressions
Just like real number fractions can be simplified, rational expressions can be
simplified.
To simplify a rational expression:
1. Factor the numerator and denominator completely.
2. Cancel any common factors in pairs – one from the numerator and one from the
denominator.
NOTE: You can only cancel factors!!!!!!!
You can perform a partial check of your answer by evaluating the original
expression and
the simplified expression. To do this, pick a value for the variable and
substitute it into
both expressions and evaluate. If they are equal you can be fairly sure that the
simplification is correct, but not absolutely sure. If you want to be more sure,
evaluate
again with another number. (Note: Don’t use restricted values and avoid the use
of 1 or
0.)
Factors That are Opposites
Sometimes factors appear to be completely different when they are just opposites
of each
other. For example, x − 2 and 2 − x appear to be different. But 2 − x = −1(x −
2). So we
can use this fact to al low further simplification when opposites occur in the
numerator
and denominator.
Section 6.2 Multiplication and Division
Recall, that when we multiply fractions we multiply the numerators together and
the
denominators together. Then, we simplify the result when possible. When we
multiply
algebraic fractions, we multiply them in essentially the same manner.
To multiply rational expressions:
1. Multiply the numerators together and the denominators together. To do this,
place
parentheses around each numerator and each denominator and “multiply”.
Example:
2. Factor A, B, C, and D completely.
3. Cancel any common factors in pairs – one from the numerator and one from the
denominator, if any exist.
4. Do not re-multiply any remaining factors.
Recall, that when we divide a fraction by another fraction, we “flipped” the
divisor then
multiplied. When we divide algebraic fractions, we divide them in essentially
the same
manner.
To divide rational expressions:
1. Place parentheses around each numerator and each denominator and “divide”.
Example:
2. Factor A, B, C, and D completely.
3. Cancel any common factors in pairs – one from the numerator and one from the
denominator, if any exist.
4. Do not re-multiply any remaining factors.
Section 6.3 Addition, Subtraction, and Least Common
Denominators
Recall that we can add or subtract fractions as long as the denominators of the
fractions
are the same. This rule holds for algebraic fractions as well.
To add rational expressions with a common denominator:
• Add the numerators. Example:
• Simplify by factoring the numerator and denominator completely, then canceling
any common factors that occur.
To subtract rational expressions with a common denominator:
• Subtract the numerators. Example:
• Simplify by factoring the numerator and denominator completely, then canceling
any common factors that occur.
Recall that when we add or subtract fractions with different denominators we
must
rewrite the fractions with common denominators. This fact is also true when we
add or
subtract algebraic fractions. When we do this, it is convention to use the
least common
denominator (LCD), which is the least common multiple ( LCM ) of the
denominators.
To find the LCD for a set of rational expressions:
• Factor each denominator completely.
• Write a product consisting of each different factor, using the greatest
exponent
that occurs for each. This product is the LCD for the set of rational
expressions.
Once you find the LCD for a set of rational expressions, you can rewrite each
expression
so that they have the same denominator. This allows us to add or subtract
algebraic
fractions.
Section 6.4 Addition and Subtraction with Unlike
Denominators
Recall that when we add or subtract fractions with different denominators we
must
rewrite the fractions with common denominators. This fact is also true when we
add or
subtract algebraic fractions.
To Add or Subtract Algebraic Fractions with Different Denominators
1. Find the LCD of all the fractions in the expression
2. Multiply each algebraic fraction by a form of 1 made up of factors of the LCD
that are missing from each fraction’s denominator. (Note: leave the Denominators
in factored form.)
3. Add or subtract the numerators, as indicated, using the distributive property
when
necessary. Write the sum or difference over the LCD
4. Factor the numerator to simplify the fraction, if possible.
Section 6.5 Complex Rational Expressions
A complex rational expression, or complex fractional expression, is a
rational
expression that has one or more rational expressions within its numerator and/or
denominator.
There are two different methods one can employ to simplify complex rational
expressions.
Simplification Method 1:Use Division
1. Simplify the numerator, by adding or subtracting the fractions in the
numerator, as
indicated
2. Simplify the denominator, by adding or subtracting the fractions in the
denominator, as indicated
3. Divide the numerator by the denominator by inverting the denominator and
multiplying.
4. Simplify if possible
Simplification Method 2: Multiplication by the LCD
1. Find the LCD of all rational expressions within the complex rational
expression.
2. Multiply the complex rational expression by a factor equal to one where this
factor is the LCD/LCD.
3. Distribute and simplify. No fractional expressions should remain within the
complex rational expression.
4. Factor the numerator and denominator, if possible, and simplify.
Section 6.6 Solving Rational Expressions
A rational, or fractional, equation is an equation containing one
or more rational
expressions.
To solve a rational equation in one variable algebraically:
1. Determine the restricted values for the equation by setting each denominator
equal to zero and solving for the variables.
2. Determine the LCD for all of the terms in the equation.
3. Clear the equation of fractions by multiplying every term in the equation by
the
LCD.
4. Solve the resulting equation.
5. Discard any solutions that are restricted values (called extraneous
solutions).
6. Check your solution(s) by substitution.
Whenever you solve an equation algebraically, one of three possibilities will
occur:
• One or more solutions exist.
• No solution exists.
• An infinite number of solutions exist. In this case the solution set is all
real
numbers except for any restricted values.
Section 6.7 Applications Using Rational Equations and
Proportions
There are many interesting applications involving rational equations. Several
are outlined
below.
The Work Principle
Suppose that A requires a units of time to complete a task and B requires b
units of time
to complete the same task. Then
• A works at a rate of
tasks per unit time,
• B works at a rate of 
tasks per unit time,
and
• A and B together work at a rate of 
tasks
per unit time.
If A and B are working together, require t units of time to complete the task.
Then both of
the following equations are true.
and
Problems Involving Motion
Problems dealing with distance, speed (or rate of travel), and time are called
distance
problems. There are three equivalent equations which we can choose from to solve
these
types of problems.
and
Problems Involving Proportions
A ratio is the quotient or comparison of two quantities with the same
unit of
measurement. A rate is the quotient or comparison of two quantities with
different units
of measurement.
Definition
Let and
be any two ratios or rates. Then the equation
is a proportion.
Thus a proportion is an equation that states that two ratios or rates are equal
to each other.
One application of proportions is in the area of geometry, especially when
studying
similar triangles.
Similar objects are objects with the same shape but
not the same size. For example:
consider a model B-52 airplane. It has the same shape as an actual B-52 airplane
but it is
much smaller in size.
Similar objects have corresponding parts. For example: an engine on the model
airplane
corresponds to an engine on the actual airplane. The relationship between the
sizes of
corresponding parts can be written as a ratio and each ratio will be the same.
Since the ratios of corresponding parts are equal, proportions can be formed.
Properties of Similar Triangles
• The ratios of corresponding sides are equal.
• The ratio of corresponding heights is equal to the ratio of corresponding
sides.
• All corresponding angles are equal.
Section 6.8 Formulas and Equations
During this semester we have learned the techniques to solve three types of
equations
• Linear equations
• Polynomial equations, and
• Rational equations.
When asked to solve an equation you should ask yourself, “What type of equation
is
this?” before you begin to try and solve it. Why? Because different techniques
are
required for each type of equation.
We can apply these techniques when asked to solve a formula for a specific
letter.
Remember that solving for a letter involves “rearranging” the other quantities
in the
equation. For example, let’s say that we want to solve for the letter x in a
given formula.
We want to rearrange the formula to be in form x = expression, where the
expression on
the right-hand side of the equation is obtained from the rearrangement of the
original
formula and does not contain the variable x.
This “rearrangement” of the other quantities must follow the guidelines set
forth for
solving the type of equation we are working with.
