4.1: Equivalent and Rational Expressions

Transcription

1 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?

2 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 as restrictions 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.

3 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

4 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:

5 Ex: Simplify the following rational expression: Ex: Simplify the following rational expression:

6 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

7 Example: Simplify the following, state the non-permissible values: Example: Simplify the following, state the non-permissible values

8 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.

9 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

10 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:

11 Example: Simplify the following sum: Example: Simplify the following difference:

12 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.

13 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"

14 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?