Rational Expressions - Cengage Learning

Teaching Guide Simplifying Rational Expressions

Preparing for Your Class Common Vocabulary • Rational expression, simplified rational expression, opposite polynomials Instruction Tips • A rational number is the ratio of two integers. A rational expression is the ratio of two polynomials. • When students evaluate rational expressions for a specified value using a calculator, they often forget to insert the parentheses that are necessary to group the numerator and denominator. For example,

•

•

•

2+6 must be entered as ( 2 + 6 ) / ( 3 − 2 ) , which is 8. If students 3−2

only enter 2 + 6 / 3 − 2 , their calculator will give them a value of 2. This is a good place to remind students that fraction bars (like parentheses) also act as grouping symbols, separating the numerator and denominators into their own groups for evaluation. Students begin the study of rational expressions with all sorts of interesting misconceptions. The most dangerous misconception is the idea that terms can be “cancelled.” For example, x +1 students are often quite sure that the x 's in can be “cancelled.” Here is a good x+2 2 +1 3 1 counterexample (let x = 2 ): . Clearly, ≠ . Where does this misconception come 2+2 4 2 from? Well, we did just teach students exponent rules not too long ago, and in those rules, 3⋅ x 1 with . Your students we taught them to replace the x 's in an expression like this: 1 4⋅ x may not have made a clear distinction between rules that apply to terms in addition and rules that apply to factors in multiplication. If this is the case, then they do not see the difference 3+ x 3x and . between expressions like 4+ x 4x Since we generally use slash marks to cross out factors that we replace with ones, students can become easily confused if they also use slash marks in situations like this: 3 − x + x + 2 . Students may not realize that in one case, the like terms sum to zero, and in the other case, 1 common factors are replaced by . It may help to use paired examples to demonstrate the 1 two concepts and differentiate between what the students see as two types of “cancelling”: 3( x − 2) . 3 + x − x + 2 vs. 2 ( x − 2) 1 and x + 2 . Consider x+2 x+2 1 are equivalent expressions and these look very similar to and that x + 2 and 1 x+2 x + 2 . Make sure to keep revisiting examples that simplify to have 1 in the numerator or 1 in

Many students do not see the difference between expressions like

the denominator so that you can find and correct any students who are making this mistake.

Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 2011, Cengage Learning

•

Even after successfully completing all the factoring and simplifying steps, students will often take incorrect action when confronted by the answer. For example, the following student x 2 + 9 x + 20 ( x + 4 ) ( x + 5 ) x +4 4 action would not be surprising: 2 = = − . If you tell = 3 x + 2 x − 15 ( x − 3) ( x + 5 ) x −3 1

1

x+4 students that the problem is done at the step, you do not give them the opportunity to x−3

make the “cancelling” mistake that they are likely to make on a test. However, you can give them this opportunity (and correct their thinking) by simply asking “Are we done?” after every step of the simplification. •

As students factor the numerator and denominator of rational expressions, it is quite helpful to write any non-factoring expressions within parentheses. For example, to simplify x 2 + 3x + 2 , we first write it with the numerator factored and the denominator in parentheses x +1 ( x + 2 )( x + 1)

as

( x + 1)

. This serves three purposes. First, it reminds students that they did try to

factor the denominator and that it did not factor. Second, it is visually easier to pick up on the factors that simplify if they look more similar. Third, students are less likely to try to “cancel” pieces of the factor if they can see it must be held together as a group. •

Unfortunately, students get so caught up in factoring trinomials of the form x 2 + bx + c , that they are suddenly flummoxed when presented with an expression like x 2 + 5 x to factor. Many will even start this by writing a set of empty binomial parentheses. Here is an example of what you are likely to see:

( )( ) ???? x2 + 5x = 2 x + 12 x + 35 ( x + 5 )( x + 7 )

So, even though GCFs do not frequently show up in these problems, you should still start every factoring process by reminding students to look for a GCF first. •

If students only see polynomial opposites written in the opposite order, like

5− x or x−5

x 2 + 3x − 4 , they can easily leap to the wrong “logical” conclusion. That is, the student 4 − 3x − x 2

begins to believe that any two expressions, written in the opposite order, are opposites. What, then, will they do with an expression like

x+3 ? It is likely that these students will 3+ x

“see” two opposites here. Make sure you change the order in some of your examples of opposites, for example

5− x 5− x is an equivalent expression of . x−5 −5 + x


Teaching Your Class Guided Learning Activity: Undefined Expressions. Using this worksheet, you can cover the topics of evaluating rational expressions (the students will need to use some scratch paper for this one) and finding numbers that cause a rational expression to be undefined. The same three expressions are used throughout the worksheet so that you can draw connections between the answers found in each variation of the procedures. (RAT-1)

Simplify rational expressions • Start with a few examples from fractions. 1

1

2⋅5 2 ⋅5 1 = = = 2 ⋅3⋅5 2 1 ⋅3⋅ 5 1 3 5+5 o What if it was written like this: ? 5 + 25 10 o Simplify: 30

1

2⋅2⋅2⋅7 2⋅2⋅2⋅ 7 8 = = 7⋅7 7 7⋅ 71 49 + 7 o What if it was written like this: ? 49

o Simplify:

56 49

=

1

1

1

1

1

1

2 ⋅ 5 ⋅ x ⋅x⋅ y ⋅ y ⋅ y 10 x 2 y 3 2⋅5⋅ x ⋅ x ⋅ y ⋅ y ⋅ y x = = o Simplify: = 4 2 ⋅3⋅5⋅ x ⋅ y ⋅ y ⋅ y ⋅ y 2 1 ⋅3⋅ 5 1 ⋅ x 1 ⋅ y 1 ⋅ y 1 ⋅ y 1 ⋅ y 3y 30 xy

•

In numerical fractions, the numerator and denominator must be factored before we simplify the fraction. The same rule applies to rational expressions. To simplify a rational expression you must remember one step: FACTOR FIRST!

Examples: Simplify.

( x + 4 )( x + 5) = x + 4 Use basic trinomial factoring. ( x − 3)( x + 5 ) x − 3

•

x 2 + 9 x + 20 x 2 + 2 x − 15

=

•

2 x2 − 8 x 2 + 7 x + 10

=

•

2 x 2 − 11x − 21 x 2 − 14 x + 49

=

( 2 x + 3)( x − 7 ) = 2 x + 3 Factor a (harder) perfect square trinomial. ( x − 7 )( x − 7 ) x − 7

•

x2 − 9 x2 + 8x − 9

=

( x + 3)( x − 3) ( x + 9 ) ( x − 1)

•

x2 + 5x x 2 + 12 x + 35

=

2 ( x + 2 )( x − 2 )

( x + 5)( x + 2 )

x ( x + 5)

( x + 5)( x + 7 )

=

=

2x − 4 Factor out a GCF and difference of squares. x+5

This does not simplify, but is tempting for students to try! x Students have difficulty with the numerator. x+7


Student Activity: Match Up on Simplifying Rational Expressions. This activity works well in groups or with students working in pairs at a whiteboard (have students draw an empty grid on the board for their answers). (RAT-3)

Why are opposite polynomials so nice? −7 7 ( x + 2)

2−5 5−2 x+4 −x − 4

3 5+3 −3 −5 − 3 x−5 (factor out a −1 ) 5− x

•

Evaluate these numeric expressions.

•

How could we simplify these?

•

For two polynomial expressions to be opposites, every corresponding pair of terms must be opposite.

•

2 − 3x + x 2 Example: 2 x + 3x − 2

− ( x + 2)

…can be rewritten as

2 + ( −3x ) + x 2

x 2 + 3 x + ( −2 )

.

2 is positive in the numerator and negative in the denominator (opposites) 3x is negative in the numerator and positive in the denominator (opposites) x 2 is positive in the numerator and denominator (not opposites) Thus, 2 − 3x + x 2 and x 2 + 3x − 2 are not opposite polynomials. 2 − 3x + x 2 could not be simplified without factoring first. x 2 + 3x − 2

•

Example:

15 − 2 x 2 x − 15


15 + ( −2 x )

2 x + ( −15 )

.

15 is positive in the numerator and negative in the denominator (opposites) 2x is negative in the numerator and positive in the denominator (opposites) Thus, 15 − 2x and 2 x − 15 are opposite polynomials. 15 − 2 x could be simplified to be −1. 2 x − 15

•

Example:

a −8 −a + 8


a + ( −8 ) −a + 8

.

a is positive in the numerator and negative in the denominator (opposites) 8 is negative in the numerator and positive in the denominator (opposites) Thus, a − 8 and −a + 8 are opposite polynomials. a −8 could be simplified to be −1 . −a + 8

•

x 2 + x − 20 Simplify. 16 − x 2

−1

( x − 4 )( x + 5) = ( x − 4 ) ( x + 5) = − ( x + 5) = − x − 5 = ( 4 − x )( 4 + x ) ( 4 − x ) 1 ( 4 + x ) ( 4 + x ) x + 4

or −


x+5 x+4

Student Activity: The Ones Recycling Center. There are all sorts of student misconceptions about what kind of expressions are really opposites and what kinds of expressions are really equivalent. This activity targets those ideas by having students work only with these concepts. (RAT-4)

Student Activity: Which of These is Not Like the Others? Students have a lot of trouble deciding on whether their answer is equivalent to the one in the book. (RAT-5)


Guided Learning Activity Undefined Expressions A rational expression is an expression of the form

A B

where A and B are polynomials and B does not equal 0. To evaluate a rational expression, you may find it helpful to first create the parentheses skeleton. Example 1: Evaluate

x2 + 4x − 5 for x = 2 and x = −3 . x2 − 9

( )2 + 4 ( ) − 5 Parentheses skeleton: ( )2 − 9 ( 2 )2 + 4 ( 2 ) − 5 4 + 8 − 5 7 Evaluate for x = 2 : = = −5 4−9 ( 2 )2 − 9 ( −3)2 + 4 ( −3) − 5 9 − 12 − 5 −8 Evaluate for x = −3: = = = undefined 9−9 0 ( −3)2 − 9 Expression

Parentheses Skeleton

You just stand there with that blank expression. Sometimes I seriously wonder if you are even trying to be rational...

Evaluate the expression for … x =1

x=4

x=0

x = −2

x = 13

3x − 12 x2 + 2x 3x 2 + 8 x − 3 x2 − 5x + 4

x 2 − 16 x2 + 2x

It is often easier to perform the evaluation if the rational expression is factored first. This way, you can quickly see the values that create a factor of zero in the numerator or denominator. Example 2:

Evaluate

x2 + 4 x − 5 for x = 0, x = 1 , and x = 3 . x2 − 9 ( x + 5)( x − 1)

Factored form:

.

( x + 3)( x − 3) ( 0 + 5)( 0 − 1) ( 5)( −1) −5 5 Evaluate for x = 0 : = = = ( 0 + 3)( 0 − 3) ( 3)( −3) −9 9 (1 + 5 )(1 − 1) = ( 6 )( 0 ) = 0 = 0 Evaluate for x = 1: (1 + 3)(1 − 3) ( 4 )( −2 ) −8 ( 3 + 5 )( 3 − 1) = (8)( 2 ) = 16 = undefined Evaluate for x = 3: ( 3 + 3)( 3 − 3) ( 6 )( 0 ) 0 As soon as you see a factor of zero in the numerator or denominator, you can quickly find the value of the answer. Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 2011, Cengage Learning

Expression

Factored Form of the Expression

Evaluate the expression for … x =1

x=4

x=0

x = 13

x = −2

3x − 12 x2 + 2 x 3x 2 + 8 x − 3 x2 − 5x + 4 x 2 − 16 x2 + 2 x

Recall that we can solve an equation like ( x + 3)( x − 5 ) = 0 using the zero property. Because of the Zero Factor Property, either x + 3 = 0 or x − 5 = 0 . This leads us to solutions of −3 and 5. Example 3:

Where is the expression

x2 + 4x − 5 undefined? x2 − 9

This expression is undefined when the denominator is equal to zero. We can answer the question by solving the equation x 2 − 9 = 0 . Factor first: ( x + 3)( x − 3) = 0 Set each factor equal to zero and solve: OR x+3=0 x−3=0 x = −3

x=3

x + 4x − 5 is undefined for −3 and 3. x2 − 9 2

The expression

Expression

Factored Form of the Expression

Set the denominator = 0. Solve the resulting equation.

Where is the expression undefined?

3x − 12 x2 + 2 x

3x 2 + 8 x − 3 x2 − 5x + 4

x 2 − 16 x2 + 2 x


Student Activity Match Up on Simplifying Rational Expressions Directions: Match each of the expressions in the squares in the table below with an equivalent simplified expression from the top. If an equivalent expression is not found among the choices A through E, then choose F (none of these). nal Ratiosions

A 1

B −1

C x+5

x D 3

E 3x

F None of these

s

Expre

9 x 3 + 15 x 3x 2 + 5

x 2 − 25 x−5

x −1 1− x

x2 + x 3x + 3

( 3x + 2 )( x + 1) 3x 2 + 5 x + 2

x −1 1+ x

3x3 − 6 x 2 x−2

3x + 1 1 + 3x

3 x3 − 27 x ( x + 3)( x − 3)

x3 + 2 x 2 + x 3x 2 + 6 x + 3

x 2 + 10 x + 25 x+5

3x − 1 1 − 3x

x2 + 6x + 5 x +1

x 2 + 25 x 2 − 25

x −8 −x + 8

18 x 2 − 3x −1 + 6 x

I can’t believe I’m saying this, but I really miss the good old days when fractions only involved numbers ...


Student Activity The Ones Recycling Center Directions: In each pair of expressions, make a decision about whether the expression is equivalent to 1 or −1. Then sort each expression into the correct recycling bin below. If an expression does not belong in either recycling bin, just leave it out! The first one has been done for you. The 1 Bin: The numerator and denominator are equivalent. The -1 Bin: The numerator and denominator are opposites.

x −1 1− x

x+2 2+ x

x−3 x+3

x2 − 1 1 − x2

x2 + 6x + 9

3( x + 4 ) ( x + 4 )( 3)

v2 − 1 v2 + 1

−x − 3 x+3

−x + 3 3− x

y2 + 1 1 + y2

x2 + 2 x − 1 x2 + 2x + 1

x2 − 2x − 3 3 + 2 x − x2

a−b −a + b

x 2 + 3x + 1 3x + 1 + x 2

5− x 5− x

( x + 3)

( x + 3)

2

2

x2 + 9

x −1 1− x


Student Activity Which of These is Not Like the Others? Directions: When you look at your answer to a problem and compare it to the answer in the back of the book or a friend’s answer, you might find that they are not quite the same. This does not mean that one of them is really different though. In each row of the table, all the expressions are equivalent except for one of them. Circle the expressions that are the same and place an X over the “oddball” expression in each row. The first one has been done for you. Here’s two hints if you’re really stuck: • You could evaluate all the expressions for the same given value and see which expressions have the same result. • You could try factoring out a −1 if the numerator and denominator look suspiciously like they might be opposites. −4 −x

1.

2.

−

3.

4.

5.

6.

−

x+3 x−4

−4 x

4 −x

−x − 3 x−4

−x − 3 −x + 4

x+3 4− x

x−5 x−5

x−5 −x + 5

x−5 5− x

x−5 x−5

−x − 4 −x − 4

x+4 x+4

x+4 4+ x

x +1 1− x

1+ x x −1

x +1 x −1

x−4 x + 4x + 3

4− x x + 4x + 3

−

x−4 ( x + 1)( x + 3)

2

−

2

−

−

4 x

−x − 4 x+4

1+ x 1− x

−

4− x x + 4x + 3 2
