Worksheet 24 (5.1)
Chapter 5 Rational Expressions
5.1 Simplifying Rational Expressions
Summary 1:
Definitions and General Properties of
Rational Numbers and Rational Expressions
A rational number can be written as a quotient of two integers, in the form , where the denominator, b, is not 0.
A rational expression is the indicated quotient of two polynomials where the value of the denominator is assumed to be nonzero.
Sign rules of rational numbers and expressions:
1.
2.
Fundamental Principle of Fractions:
If b and k are nonzero integers and a is any integer, then.
Simplifying a rational expression:
1. Completely factor the polynomial given in the numerator and denominator.
2. Apply the fundamental principle of fractions by dividing the common factor or factors.
3. The simplest form will be the quotient of the product of remaining factors in the numerator and the product of remaining factors in the denominator.
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Worksheet 24 (5.1)
Warm-up 1. Reduce to lowest terms:
a)=
=
b)=
=
c)=
=
d) =
=
=
Problems - Simplify:
1. 2.
3. 4.
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Worksheet 25 (5.2)
5.2 Multiplying and Dividing Rational Expressions
Summary 1:
Multiplying Rational Expressions
Basic definition for multiplying rational numbers:
If a, b, c, and d are integers with b and d not equal to zero, then.
Multiplying rational expressions:
1. Completely factor each numerator and denominator.
2. Apply the basic definition for multiplying rational numbers by rewriting the numerator as a product of factors and rewrite the denominator as a product of factors.
3. Simplify by dividing common factors.
4. The result is the quotient of the product of remaining factors in the numerator and the product of remaining factors in the denominator.
Warm-up 1. Multiply and simplify:
a)=
=
b)=
=
c)=
=
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Worksheet 25 (5.2)
Problems - Perform the indicated operation. Express in simplest form:
1.
2.
3.
Summary 2:
Dividing Rational Expressions
Basic definition for dividing rational numbers:
If a, b, c, and d are integers with b, c, and d not equal to zero, then .
Note:are called reciprocals or multiplicative inverses.
Dividing rational expressions:
1. Apply the basic definition for dividing rational numbers.
2. Follow the steps for multiplying rational expressions in summary 1.
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Worksheet 25 (5.2)
Warm-up 2. Divide and simplify:
a)=
=
b)=
=
=
c)=
=
=
Problems - Perform the indicated operation. Express in simplest form:
4.
5.
6.
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Worksheet 26 (5.3)
5.3 Adding and Subtracting Rational Expressions
Summary 1:
Adding Rational Expressions
Basic definition for adding rational numbers with a common denominator:
If a, b, and c are integers and b is not zero, then.
Equivalent fractions are fractions with different denominators that name the same number.
The LCD, least common denominator, is the least common multiple of a set of denominators. The LCD can be determined by inspection or by writing the product of the highest power for each factor in any given polynomial.
Adding rational expressions:
1. Determine if the given rational expressions have a common denominator.
2. If not, find the LCD.
3. Rewrite the given rational expressions as equivalent rational expressions using the LCD in the denominators.
4. Apply the basic definition for adding rational numbers with a common denominator.
5. Express result in simplest form.
Warm-up 1. Add and simplify:
a)=
=
=
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Worksheet 26 (5.3)
b)=
=
=
c)
The LCD is ______.
=
=
=
d)
The LCD is ______.
=
=
=
Problems - Add and express in simplest form:
1.
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Worksheet 26 (5.3)
12.
13.
14.
Summary 2:
Subtracting Rational Expressions
Basic definition for subtracting rational numbers with a common denominator:
If a, b, and c are integers and b is not zero, then.
Subtracting rational expressions:
1. Determine if the given rational expressions have a common denominator.
2. If not, find the LCD.
3. Rewrite the given rational expressions as equivalent rational expressions using the LCD in the denominator.
4. Apply the basic definition for subtracting rational numbers with a common denominator.
5. Express in simplest form.
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Worksheet 26 (5.3)
Warm-up 2. Subtract and simplify:
a)=
=
=
b)=
The LCD is ______.
=
=
=
c)=
The LCD is ______.
=
=
=
=
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Worksheet 26 (5.3)
Problems - Subtract and simplify:
5.
6.
7.
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Worksheet 27 (5.4)
5.4 More on Rational Expressions and Complex Fractions
Summary 1:
Complex fractions are fractional forms that contain rational numbers or rational expressions in the numerators and/or denominators.
Simplifying a complex fraction - Method A:
1. If necessary, perform the indicated operation in the numerator and/or denominator.
2. Rewrite the indicated quotient as the division of two rational numbers or rational expressions.
3. Apply the basic definition for dividing rational numbers.
Simplifying a complex fraction - Method B:
1. Find the LCD of the rational numbers or rational expressions in the given complex fraction.
2. Multiply both numerator and denominator by the LCD to obtain an equivalent fraction. (If necessary, apply the distributive property.)
3. Simplify the resulting expression.
Warm-up 1. Simplify using method A:
a)=
=
=
=
b)=
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Worksheet 27 (5.4)
=
=
=
=
Simplify using method B:
c)=
=
=
d)=
=
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Worksheet 27 (5.4)
=
=
Problems - Simplify:
1.
2.
3.
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Worksheet 28 (5.5)
5.5 Dividing Polynomials
Summary 1:
Dividing a polynomial by a monomial:
1. Divide each term of the polynomial by the common monomial divisor.
2. Reduce each fraction to its simplest form.
Dividing a polynomial by a polynomial:
1. Use the conventional long division format and arrange both the dividend and the divisor in descending order.
2. The first term of the quotient is determined by dividing the first term in the dividend by the first term of the divisor.
3. Multiply each term of the divisor by the term of the quotient found in step 2. Position this product so that it can be subtracted from the dividend.
4. Subtract.
5. Repeat the process beginning with step 2 using the resulting polynomial from step 4 as the new dividend.
Warm-up 1. Divide:
a)=
=
b)
x2 - 3 The quotient is ______.
┌────────────────
x + 1 │ x3 + x2 - 3x - 3
-( )
- 3x - 3
-( )
0 The remainder is _____.
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Worksheet 28 (5.5)
c)
2x2 + 10
┌────────────────────────
x2 - 5 │ 2x4 + 0x3 + 0x2 - 3x + 1
-( )
10x2 - 3x + 1
-( )
- 3x + 51
=
Problems - Divide:
1.
2.
3.
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Worksheet 29 (5.6)
5.6 Fractional Equations
Summary 1:
The restricted values of a given rational expression are those values that make the denominator equal to 0.
Solving equations involving rational expressions:
1. Determine the restricted values for each rational expression in the given equation.
2. Find the LCD in the equation.
3. Apply the multiplication property of equality, using the LCD, to clear the equation of all fractional forms.
4. Solve the resulting equivalent equation.
5. Check when directed to do so.
Note: If a restricted value results as a solution to the equation found in step 4, then the solution set is Æ.
Warm-up 1. Solve:
a) =
The restricted value is _____, (x ¹ ____).
The LCD is ______.
=
=
=
=
=
The solution set is { }.
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Worksheet 29 (5.6)
b) =
The restricted value is _____, (y ¹ ____).
The LCD is ______.
=
=
=
=
=
The solution set is ______.
Problems - Solve:
1.
2.
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Worksheet 29 (5.6)
Summary 2:
Ratio and Proportion
A ratio is the comparison of two numbers with division.
A proportion is a statement of equality between two ratios.
The Cross-Multiplication Property of Proportions:
Ifwhere b¹0 and d¹0, then.
Solving fractional equations as proportions:
1. Find restricted values for each rational expression in the given equation.
2. Determine if the given equation is in the form.
3. If so, apply the cross-multiplication property of proportions.
4. Solve resulting equivalent equation.
5. Check when directed to do so.
Warm-up 2. Solve:
a) =
=
=
=
= The solution set is { }.
Problem - Solve:
3.
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Worksheet 29 (5.6)
Summary 3:
Word Problems Using Fractional Equations
Follow the listed steps in Summary 3 for Section 2.4.
Warm-up 3. Set up and write an algebraic equation, then solve:
a) The sum of two numbers is 40. If the larger is divided by the smaller, the quotient is 5 and the remainder is 4. Find the numbers.
Declare the variable:
Let x = larger number
= smaller number
Write an algebraic equation and solve:
=
=
=
=
=
=
The numbers are _____ and _____.
Problem - Set up and write an algebraic equation, then solve:
4. The perimeter of a rectangle is 104 meters. If the ratio of its width to its length is 5 to 8, find the dimensions of the rectangle.
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Worksheet 30 (5.7)
5.7 More Fractional Equations and Applications
Summary 1:
The restricted values of a given rational expression are those values that make the denominator equal to 0.
Solving fractional equations:
1. Factor the denominators.
2. Determine the restricted values for each rational expression in the given equation.
3. Find the LCD in the equation.
4. Apply the multiplication property of equality, using the LCD, to clear the equation of all fractional forms.
5. Solve the resulting equivalent equation.
6. Check when directed to do so.
Note: If a restricted value results as a solution to the equation found in step 5, then the solution set is Æ.
Warm-up 1. Solve: a)
The restricted values are ___ and ___, (x ¹ ___ and x ¹ ___).
The LCD is ______.
(3x + 1) ( ) + (5x - 1) ( ) = (x + 3) ( )
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Worksheet 30 (5.7)
3x + 1 + 5x2 - 16x + 3 =
( ) = 0
( ) ( ) = 0
x = ______and x = ______
The solution set is { }
Problems - Solve:
1.
2.
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Worksheet 30 (5.7)
Summary 2:
Word Problems Using Fractional Equations
Follow the listed steps in Summary 3 for Section 2.4.
Warm-up 2. Set up and write an algebraic equation, then solve:
a) A tank can be filled by the hot-water faucet in 4 minutes and by the cold-water faucet in 6 minutes. How long does it take to fill the tank using both faucets together?
Declare the variable:
Let x = time required using both faucets
Write an algebraic equation and solve:
=
=
The amount of time it takes to fill the tank with both faucets is
______minutes.
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Worksheet 30 (5.7)
Problems - Set up and write an algebraic equation, then solve:
3. John can pick a bushel of peaches in 30 minutes. His little sister can pick a bushel of peaches in 45 minutes. How long will it take them to pick a bushel of peaches working together?
4. One pipe can fill a tank in 6 hours. The drain can empty the tank in 10 hours. How long will it take to fill the tank if the drain is left open?
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