MATH 21
Lesson Five

CHAPTER #5 - RATIONAL EXPRESSIONS (Pages 218 to 276)

LESSON #5.1 - The General objective of this lesson is to learn how to simplify rational expressions.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to express rational expressions in reduced form.

WORD PROBLEMS:

1. In Mrs. Aguilar's bread recipe, flour and water are to mixed in the ratio of 3 parts water to 5 parts flour. How many parts of flour are needed to make a mixture of 61 parts of water and flour?
   
2. An inlet pipe can fill a tank in 10 minutes.  A drain pipe can empty the same tank in 12 minutes.  If the tank is empty and both pipes are open, how long will it take before the tank overflows?

5.1 - Simplifying Rational Expressions - Rational expressions are to algebra what rational numbers are to arithmetic. This means that it is necessary to understand how to add, subtract, multiply and divide using rational expressions.

Definitions:

• Rational Number - A rational number is one that can be written as the ratio of two integers.
• Rational Expression - A rational expression is a ratio of two polynomials.

  They both obey the following properties.

• Property of Signs - Given two integers or polynomials A and B, then;

• Fundamental Principle of Fractions - Given three integers or polynomials A, B and C, then;

Board and Class Problems - Pages 223 to 224/16, 20, 34, 56, 68.

LESSON #5.2 - The General objective of this lesson is to learn how to multiply and divide rational expressions.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:

• Multiply rational expressions.
• Divide rational expressions

5.2 - Multiplying and Dividing Rational Expressions - Rational expressions are handled the same as rational numbers.

Definitions:

• Multiplication - If A, B, C and D are polynomials, with B and D not equal to zero, then -

• Division- If A, B, C and D are polynomials, with B and D not equal to zero, then -

Board and Class Problems - Pages 230 to 231/16, 22, 30, 40, 50.

LESSON #5.3 - The General objective of this lesson is to learn how to add and subtract rational expressions.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to; Add rational expressions. Subtract rational expressions

5.3 - Adding and Subtracting Rational Expressions - recall that when adding or subtracting fractions all of the denominators must be the same. Rational expressions are handled the same as rational numbers

Like Denominators - To add or subtract fractions with like denominators, you add or subtract the numerators to find the numerator of the sum or difference, and retain the common denominator.

Unlike Denominators - To add or subtract fractions with unlike denominators, change them to equivalent fractions having the same denominator (Least Common Denominator), and then follow the rule for like denominators.

Least Common Denominator - To find the LCD you follow the following procedure.

Step 1. Factor each denominator into a product of its prime factors.

Step 2. List each prime factor with the largest exponent it has in any factored denominator.

Step 3. The LCD is the product of the factors listed in Step 2.

Board and Class Problems - Pages 237 to 238/14, 28, 54, 66, 68f.

LESSON #5.4 - The General objective of this lesson is to learn how to deal with complex fractions.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to; Simplify complex fractions.

5.4 - More on Rational Expressions and Complex Fractions - So far we have only considered fractions where both the numerator and the denominator are polynomials. If a numerator and/or denominator of a rational expression contains a fraction itself, the rational expression is called a complex fraction.

Complex fraction - A complex fraction is said to be simplified when fractions have been cleared from both the numerator and the denominator, and the resulting fraction has been reduced to its lowest terms. This is done by the following the steps.

Step 1 - Combine the fractions in the numerator, and in the denominator.

Step 2 - Carry out the indicated division of the numerator by the denominator.

Board and Class Problems - Pages 247 to 248/38, 42, 44, 50, 60.

LESSON #5.5 - The General objective of this lesson is to learn how to learn how to divide using polynomials.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to divide using the long division algorithm.

5.5 -Dividing Polynomials - To divide one polynomial by another, the following techniques are available to us -

• Factoring.
• Long division algorithm.
• Synthetic division

Factoring is the easiest way of dividing one polynomial by another, but it depends upon the ability to factor the numerator, and that one of those factors is identical to the denominator. The long division algorithm does not rely upon those two conditions.

Long Division Algorithm - We can express this algorithm as follows -

Board and Class Problems-

1. Divide  5x + x² + 6  by  x - 3.
2. Divide  x³ + 2x² + 1  by  x - 1.

Arrange both the divisor and the dividend in decreasing powers of the variable. Include all powers as shown, and then divide.

The solution is -

Multiplying this answer by  x - 3  will yield the dividend.

Synthetic Division - In the division algorithm the only restriction we had for the divisor is that its degree be less than or equal to the degree of the dividend. When the divisor is of the form -

x - k

the process can be shortened considerably. This is discussed in Appendix A on pages 621 to 625. Consider the problem -

If we rewrite it and omit the powers of x we would have -

After eliminating the selected numbers (bold face) which are either duplicates or obvious, and collapsing the system, we are left with -

If we change the sign in the divisor and add instead of subtracting this becomes -

Since we have arranged the dividend in decreasing order of degree, and we are dividing by a first-degree term, the degree of the quotient must be one less than the degree of the dividend. We now have the coefficients of the quotient, including the remainder.

Board and Class Problem - Divide  x⁴ + 2x² + 1  by  x - 1.

LESSON #5.6 - The General objective of this lesson is to expand our study of fractional equations.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve fractional equations containing the variable in the denominator.

WORD PROBLEM - Contained within the lesson..

5.6: Fractional Equations - Any equation containing one or more fractions can be classified as a fractional equation. Usually the first step to be carried out is to multiply both sides of the equation by the least common denominator (LCD). When the variable occurs in any of the denominators, care must be exercised to exclude any "apparent solutions" that makes a denominator equal to zero.

Procedure for Solving Fractional Equations -

Step 1. Determine the LCD for the fractions in the equation.

Step 2. Multiply both sides of the equation by the LCD.

Step 3. Solve the resulting equation for the unknown.

Step 4. Substitute the solution from Step 3 into all of the denominators to ensure that no denominator equals zero. If any denominator equals zero, reject the 'apparent solution.'

Board and Class Problems - Pages 260 to 261/28, 36, 45, 58.

5.7: More Fractional Equations and Applications -

Board and Class Problems - Pages 269 to 271/14, 28, 47, 54.

HOMEWORK - Lesson Five Assignment

COMMENTS -

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