Math 3201
Chapter 4: Rational Expressions and
Equations
12 Hours
4.1: Equivalent and Rational Expressions
A Rational Expression is any expression that can be written as the quotient of two
polynomials, in the form ­
, where Q(x) ≠ 0.
☆ A fraction that has a polynomial for either the numerator, denominator
or both .
☆ A fraction with variables in it
☆All rational expressions are algebraic functions but not all algebraic
functions are rational expressions.
Which of the following are rational expressions?
Non-Permissible Values (NPV):
☆values of a variable that make the denominator of a rational expression
equal 0.
☆To find the NPV, let the denominator equal zero and solve for the
variable. You may need to factor in order to solve the equation.
☆ All NPVs of a rational expression must be stated asrestrictions on the
variable to ensure that the expression is defined.
Ex: Determine the non-permissible value(s) for each rational expression, and then
state all restrictions. Verify your answer!
a)
b)
Non-Permissible Values vs. Inadmissible Values
☆ Non-permissible values and inadmissible values are different.
☆ Inadmissible values are values that do not make sense in a given
context. (Ex: Negative speed)
☆ Non-permissible values are values that cause the denominator of a
rational expression to equal zero.
Creating Equivalent Rational Expressions
Recall: Given the fraction
, multiply both the numerator and
denominator by 3, by 5 and by -4.
☆ Does the value of the rational number change when you perform this
operation?
We just created equivalent rational numbers. If we multiply the
numerator and denominator of a rational expression by the same thing, do
we create equivalent expressions?
Equivalent Rational Expressions?? Or Not??
Handout
RULE:
☆If the expressions are not equal for the same value of x,
then the expressions are not equivalent.
☆ If you introduced a factor that required a new restriction,
other than the original restriction, then the expressions would
not be equivalent.
4.1 Exit Card!!
4.2: Simplifying Rational Expressions
☆ The benefit of simplifying an expression is to create an equivalent
expression that is easier to evaluate.
☆ The simplified expression MUST retain the non-permissible values of the
original expression for both to be equivalent.
Ex: Simplify the following rational expression:
Ex: Simplify the following rational expression:
Ex: Simplify the following rational expression:
Error Analysis
Centres!!
4.3: Multiplying & Dividing Rational Expressions
Multiplying Rational Expressions:
☆ Identify the non-permissible values of the variable
☆ Factor the numerators and denominators of both expressions, where
possible
☆ Multiply the numerators and multiply the denominators, writing each
product as a single rational expression.
☆ Simplify using common factors
☆ Write the product, stating the restrictions on the variable
Example: Simplify the following, state the non-permissible values:
Example: Simplify the following, state the non-permissible values
Dividing Rational Expressions:
☆ Identify the non-permissible values of the variable
☆ Remember to consider both the numerator and the denominator of the
divisor. This ensures that the second rational expression (or divisor) is not
equal to zero.
☆ Factor the numerators and denominators, where possible
☆ Multiply by the reciprocal
☆ Simplify using common factors
☆ Write the quotient and state the restrictions
Example: Simplify the quotient.
Example: Simplify the quotient.
4.4: Adding and Subtracting Rational Expressions
Adding or Subtracting Rational Expressions:
☆ Factor the numerators and denominators of both expressions, if possible
☆ Determine the lowest common denominator (LCD)
☆ Rewrite each rational expression as an equivalent expression with the LCD
as the denominator
☆ Add or subtract the numerators of the equivalent expressions
☆ Simplify the rational expression and restate the restrictions on the
variable
The strategies for finding the LCD with rational expressions are the same
strategies for finding the LCD for fractions. There are four situations to
consider:
Example: Simplify the following sum:
Example: Simplify the following difference:
Example: Simplify the following expression:
4.5: Solving Rational Equations
☆ You can solve a rational equation algebraically by multiplying each term in the
equation by the LCD to eliminate fractions from the equation.
☆ Solve the resulting linear or quadratic equation
☆ If a root of a rational equation is a non-permissible value of the rational
expressions in the original equation, it is an extraneous root and must be
dismissed as a valid solution.
☆ Each solution to a rational equation can be verified by substituting it into the
original equation. If the left side equals the right side, then it is a valid
solution.
Example: Solve the following equation for x:
Steps for Solving Word Problems:
☆ Read the problem carefully and understand what is being asked
☆ Introduce a variable to represent an unknown quantity
☆ Write an algebraic equation to represent the given information
☆ Solve the equation
☆ State the solution to the problem. Check that the solution "makes sense"
Example: When they work together, Stuart and Lucy can deliver flyers to all the
homes in their neighbourhood in 42 min. When Lucy works alone, she can deliver
the flyers in 13 min less time than Stuart can when he works alone. When
Stuart works alone, how long does he take to deliver the flyers?
Example: Rita bought a case of concert T-shirts for $450. She kept two
T-shirts for herself and sold the rest for $560, making a profit of $10 on each
T-shirt. How many T-shirts were in the case?