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Chapter 7 Rational Expressions
Algebra II Common Core
Lesson 1: Intro to Rational Functions and Undefined Values Lesson 2: Simplifying Rational Expressions Lesson 3: Multiplying and Dividing Rational Expressions Lesson 4: Adding and Subtracting Rational Expressions Lesson 4 Practice: Adding & Subtracting Rational Expressions Lesson 5: Complex Fractions Lesson 6: Fractional Equations
This assignment is a teacher-modified version of Algebra 2 Common Core Copyright (c) 2016 eMath Instruction, LLC used by permission.
UNIT 7 LESSON 1
INTRODUCTION TO RATIONAL FUNCTIONS
Rational functions are simply the __________________ of polynomial functions. They take on more interesting properties and have more interesting graphs than polynomials because of the interaction between the numerator and denominator of the fraction.
Recall: Compositions of functions
Exercise 1: If g(x) = 3x – 2 and
then find:
(a) f(g(-2)) (b) f(g(2)) (c) f(g(x))
Undefined: Is when the denominator of a fraction is equal to ________________. A fraction
cannot have zero in the denominator because we cannot divide by ________________.
Exercise 2: Consider the rational function given by
.
(a) Algebraically determine the y-intercept for this
function.
(b) Algebraically determine the x-intercept of this
function. Hint – a fraction can only equal zero
if its numerator is zero.
(c) For what value of x is this function undefined? Why is it undefined at this value? (d) Based on (c), state the domain of this function in set-builder notation. (e) Sketch a graph of this function.
Domain of Rational Expressions:
The domain is the set of values that x can be in an expression. Since, we cannot divide by zero, the domain of rational expressions is _______________________________________________ ________________________________ (where the function is undefined).
Exercise 3: Find the domain of the following rational expressions.
(a)
(b)
Exercise 4: Find all values of r for which the rational function f(r) =
is undefined.
Exercise 5: Which of the following represents the domain of the function
?
(1) {x| x ≠ ±4} (3) {x| x≠-2 and 8}
(2) {x| x ≠ 3} (4) {x| x ≠ -6 and 3}
Exercise 6: What is the domain of the function
?
Exercise 7: State the values of x that are not in the domain of the function f(x) =
.
Exercise 8: What is the domain of the function
? Justify.
Exercise 9: Which function has a greater Average Rate of Change over the interval [2,7], f(x) or g(x)? Justify.
INTRODUCTION TO RATIONAL FUNCTIONS CC ALGEBRA II HOMEWORK LESSON 1
FLUENCY
1. Which of the following values of x is not in the domain of
?
(1) x = -7 (3) x = 3
(2) x = 7 (4) x = -3
2. Which of the following values of x is not in the domain of
?
(1)
(3)
(2) x = -1 (4) x = -3
3. Which values of x, when substituted into the function
, would make it
undefined?
(1) x = 2 and 8 (3) x = -4 and 4
(2) x = -4 and 8 (4) x = -4 and 0
4. Which of the following represents the domain of
?
(1) {x | x ≠ ±2} (3) {x | x ≠ -4 and 14}
(2) {x| x ≠ -7 and 2} (4) {x | x ≠ -5 and 14}
5. Which of the following represents the domain of
?
(1) {x | x ≠ 1/3} (3) {x | x ≠ -1/2 and 5}
(2) {x | x ≠ -1/3 and ½} (4) {x | x ≠ -2 and 5/2}
6. If f(x) = 2x + 7 and
then g(f(-5)) = ?
(1) -1 (3) 6
(2) 2 (4) -3
7. If
and g(x) = 4x – 1 then f(g(x)) = ?
(1)
(3)
(2)
(4)
8. The y-intercept of the rational function
is
(1) 15 (3) -3
(2) -5 (4) 12
9. Determine where the function is undefined.
(a)
(b)
LESSON 2
SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II
Simplifying a rational expression into its lowest terms is an extremely useful skill. Every time we simplify a fraction, we are essentially finding all ______________________ of the numerator and denominator and dividing them to be equal to one. The numerator and denominator must be __________________ and only ___________________ factors cancel each other. This is true whether our fraction contains monomial, binomial, or polynomial expressions.
Exercise #1: Simplify each of the following monomials dividing other monomials.
(a)
(b)
(c)
Exercise #2: Which of the following is equivalent to
?
(1)
(3)
(2)
(4)
When simplifying rational expressions that are more complex, always __________________, then identify common factors that can be eliminated. Exercise #3: Simplify each of the following rational expressions.
(a)
(b)
(c)
A special type of simplifying occurs whenever expressions of the form (x – y) and (y – x) are involved. Exercise #4: Simplify each of the following fractions.
(a)
(b)
(c)
(d) What do you notice?
Exercise #5: Which of the following is equivalent to
?
(1)
(3)
(2)
(4)
Exercise #6: Which of the following is equivalent to
?
(1) –
(3)
(2)
(4)
Exercise #7: Simplify each rational expression below.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
SIMPLIFYING RATIONAL EXPRESSIONS CC ALGEBRA II HOMEWORK LESSON 2
FLUENCY
1. Write each of the following ratios in simplest form.
(a)
(b)
(c)
2. Which of the following is equivalent to the expression
?
(1)
(3)
(2)
(4)
3. Simplify each of the following rational expressions.
(a)
(b)
(c)
(d)
(e)
(f)
4. Which of the following is equivalent to the fraction
?
(1)
(3)
(2)
(4)
–
5. The rational expression
can be equivalently rewritten as
(1)
(3)
(2)
(4)
6. Written in simplest form, the fraction
is equal to
(1) 5y – 5x (3) –
(2)
(4)
LESSON 3
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II
Multiplication of rational expressions follows the same principles as those involved in simplifying them.
Exercise #1: Simplify each of the following rational expressions.
(a)
(b)
The ability to “cross-cancel” with fractions is a result of the two facts: (1) to multiply fractions we multiply their respective numerators and denominators and (2) multiplication is commutative. The keys to multiplication are – factor and then reduce. You can only reduce numerators with denominators.
Exercise #2: Simplify each of the following products.
(a)
(b)
(c)
(d)
Since division by a fraction can always be thought of in terms of multiplying by its __________________, these problems simply involve an additional step. Remember: ________________________________________
Exercise #3: Perform each of the following division problems. Express all answers in simplest form.
(a)
(b)
(c)
(d)
Exercise #4: When
is divided by
the result is
(1)
(3)
(2) 3x - 15 (4) 9x - 5
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS CC ALGEBRA II HOMEWORK LESSON 3
SKILLS
1. Express each of the following products in simplest form.
(a)
(b)
(c)
2. Write each of the following products in simplest form.
(a)
(b)
(c)
(d)
3. When
is divided by
the result is
(1) 2x8y7 (3)
(2)
(4)
4. Express the result of each division problem below in simplest form.
(a)
(b)
(c)
(d)
LESSON 4
COMBINING RATIONAL EXPRESSIONS WITH ADDITION AND SUBTRACTION
Occasionally it will be important to be able to combine two or more rational expressions by addition. Least Common Denominator:
When doing certain rational expression problems, we need to find a least common
denominator. We use them for adding and subtracting, complex fractions and rational
equations.
To find a Least Common Denominator:
1.) ______________________ the given denominators.
2.) Take the product of all the ______________________ factors.
Each factor should be raised to a power equal to the greatest number of times that
factor appears in any one of the factored denominators.
Example 1: Find the LCD of the following rational expressions.
(a)
(b)
(c)
(d)
Guidelines for Adding & Subtracting Fractions: 1.) ______________________ each denominator ______________________.
2.) Find the ____________________ by making a list of all of the denominators. (All of your factors must be “represented”). 3.) The final LCD must be listed as ________________________. 4.) Make sure all terms have the ________________________________. Remember, the only way to keep a fraction the same value is by multiplying by 1. Therefore, you must multiply both the numerator and denominator by the ________________ term. 5.) Keep the denominator, and combine _____________________ in the numerator. 6.) Reduce, if necessary.
Exercise #2: Combine each of the following fractions by first finding a common denominator. Express your answers in simplest form.
(a)
(b)
(c)
Exercise #3: Combine each of the following fractions. Simplify.
(a)
(b)
When we subtract rational expressions, it is important to ____________________ the negative to the numerator of the fraction and change it to an addition problem. Then we follow the same process that we did before. Exercise #4: Perform each of the following subtraction problems. Express your answers in simplest form.
(a) 2 2
3 7 3
4 4
x x
x x
(b)
2
3 2
4 1 10 5
x
x x
(c) 2 2
6
4 8 20
x
x x x
(d)
2 2
2 8
5 4 12 32
x
x x x x
Exercise #5: Which of the following is equivalent to 1 1
1x x
?
(1) 1
x
x (3)
2
1
x x
(2) 2
1
x x (4)
2 1
x
x
LESSON 4 COMBINING RATIONAL EXPRESSIONS WITH ADDITION AND SUBTRACTION
COMMON CORE ALGEBRA II HOMEWORK
FLUENCY
1. Find the LCD of the following fractions.
(a)
&
(b)
&
(c)
&
(d)
2. Combine each of the following using addition. Simply you result whenever possible.
(a) 3 1 2 5
6 9
x x (b)
1
10 15
x
x
3. Combine each of the following using addition. Simplify your final answers.
(a) 2
2 3
5 25 3 40
x x
x x x
(b)
2 2
4 2
24 128 12 32
x
x x x x
4. Which of the following represents the sum of 1 1
and 1 1x x
?
(1) 2
2
1
x
x (3)
2
1x
(2) 1
x (4)
2
2
1
x
x
5. When the expressions 2
2 2
8 3 6 and
9 9
x x x
x x
are added the result can be written as
(1) 5
3
x
x
(3)
2
3
x
x
(2) 2
3
x
x
(4)
7
3
x
x
6. Express the following differences in simplest form.
2 2
2 4
4 32 16
x
x x x
7. When 7 14
3 12
x
x
is subtracted from
2 6
3 12
x
x
the result can be simplified to
(1) 5
3 (3)
10
3
(2) 2
3 (4)
7
3
LESSON 4 PRACTICE COMBINING RATIONAL EXPRESSIONS WITH ADDITION AND SUBTRACTION
COMMON CORE ALGEBRA II
More practice with adding and subtracting rational expressions! Exercise 1: Simplify each expression below.
(a)
(b)
(c)
(d)
(e)
(f)
Exercise 2: Expressed in simplest form,
is equivalent to
(1)
(2)
(3)
(4)
Exercise 3: The expression
is equivalent to
(1)
(3)
(2)
(4)
Exercise 4: Algebraically prove that
, where .
LESSON 4 PRACTICE COMBINING RATIONAL EXPRESSIONS WITH ADDITION AND SUBTRACTION
HOMEWORK
Complete the following questions. Make sure to simplify all of your final answers. 1.) Simplify each expression.
(a)
(b)
(c)
(d)
(e)
(f)
2.) Algebraically prove that
, where .
3.) Simplify:
(1)
(2)
(3)
(4)
LESSON 5 COMPLEX FRACTIONS
COMMON CORE ALGEBRA II
Complex fractions are simply defined as fractions that have __________________ within their numerators and/or denominators. To simplify these fractions means to remove these minor fractions and then eliminate any common factors. The key, as always, is to multiply by the number one in ways that simplify the fraction.
Exercise #1: Consider the complex fraction
1 1
9 181
3
.
By multiplying the major fraction by the number one, by using the ___________________________, we will always eliminate the minor fractions (by turning them into integer expressions).
1) Find the LCD (least common
multiple) of all of the minor
fractions
2) Multiply every term in
numerator and denominator by the
LCD (Cancel denominator with all
or part of the LCD, what remains
gets multiplied by the numerator)
3) Combine like terms in
numerator and denominator
4) Make sure answer is in simplest
form.
Exercise #2: Simplify each of the following complex fractions.
(a)
1 1
2 102
5
(b)
2 2
35 5
3
x
x
(c)
3 1
8 47 3
2 4
x
x
Exercise #3: Simplify each of the following complex fractions.
(a) 2
2
1 2
23 3
2
x
x x
(b)
2
2 2
51 1
5
x
x x
(c)
1 2
12 64
12 3
x
xx
x
If the denominators of the minor fractions become more complex, be sure to factor them first, just as you did with the addition and subtraction in the previous lesson.
Exercise #4: Simplify each of the following complex fractions.
(a)
2
4 2
2 412 24
2 8
x xx
x x
(b) 2
2
1
6 2
4
8 12
x
x x
x
x x
LESSON 5 COMPLEX FRACTIONS
COMMON CORE ALGEBRA II HOMEWORK
FLUENCY
1. Simplify each of the following numerical complex fractions.
(a)
1 3
4 201
2
(b)
5 1
18 61
3
2. Simplify each of the following complex fractions.
(a)
12
25
1
x
x
(b)
2
1 1
8 21 1
12 3
x
x x
3. Simplify each of the following complex fractions.
(a)
1 2
10 101
2 10
x
xx
(b)
2
33
41
28
x
x
4. Simplify each of the following complex fractions.
(a)
2
4
4 105 10
14 40
x
x xx
x x
(b) 2
2
3 2 8
1 4
2 12
5 4
x
x x
x x
x x
5. Which of the following is equivalent to
2
1 1
11
x x
x x
?
(1) 1 (3) 1
x
x
(2) 2
1x (4) 2x x
6. Express in simplest form:
LESSON 6 SOLVING FRACTIONAL EQUATIONS
COMMON CORE ALGEBRA II
Equations involving fractions or rational expressions arise frequently in mathematics. The key to working with them is to manipulate the equation, typically by multiplying both sides of it by some quantity that eliminates the fractional nature of the equation. The most common form of this practice is “cross-multiplying.”
Exercise #1: Use the technique of cross multiplication to solve each of the following equations.
(a) 4 5 1
2 5
x x (b)
1 2
2 6
x
x x
There are rational equations where we cannot cross-multiply. This occurs when we have more than one fraction on one side of the equation. We must use another method to solve these equations. Steps: 1.) Find the _________________ of all the fractions. 2.) Multiply every term by the LCD to ___________________ all of the fractions. 3.) Solve the resulting equation. 4.) Check your answer in the original equation. Reject any extraneous roots. *Any root where the denominator is equal to 0 is extraneous*
Exercise #2: Solve the following equations.
(a) 1 9 3
2 4 4x x (b)
Exercise #3: Which of the following values of x solves: 4 2 31
6 10 15
x x ?
(1) 14x (3) 8x (2) 6x (4) 11x
Exercise #4: Solve the following equation for all values of x.
2 2
1 3 1 1 1
2 4 2x x x x
Exercise #5: Solve and make sure to reject any extraneous roots.
(a) 2
1 18 9
5 8 15 3
x
x x x x
(b)
2
4 1 1
4 12 6 2
x
x x x x
Exercise #6: Which equation has a greater x-intercept, f(x) or g(x)?
LESSON 6 SOLVING FRACTIONAL EQUATIONS
COMMON CORE ALGEBRA II HOMEWORK
FLUENCY
1. Solve each of the following fractional equations.
(a) 2 1 3
3 6 2
x x (b)
5 13
2 2x
2. Solve each of the fractional equations for all value(s) of x.
(a) 12
8xx
(b) 2
3 1 1 1
4 2 2 3x x x
(c) 17 11 5 8
3 3
x
x x x
3.Solve the following equation for all values of x. Express your answers in simplest a bi form.
3
9 1
x x
x
4.Solve each of the following equations. Be sure to check for extraneous roots.
(a) 2
1 2 2
5 6 11 30
x
x x x x
