The document discusses rational expressions and operations involving them. It begins with an introduction to rational expressions, noting they are algebraic expressions with both the numerator and denominator being polynomials. It then outlines the lessons that will be covered in the module, including illustrating, simplifying, and performing operations on rational expressions. Several examples are then provided of simplifying rational expressions by factoring the numerator and denominator and cancelling common factors. The document also discusses multiplying rational expressions by using the same process as multiplying fractions, and provides examples of multiplying rational expressions.

2. Ms. Lorie Jane L Letada
Rational Expressions is an algebraic expression where both
numerator and denominator are polynomials. Duringelementaryyears,youhave
learned that a fraction is a ratio of a numerator and denominator where the
denominator is not equal to zero, but in this module, we extend the concept of
fractions to algebraic expressions.
Here is the map of the lessons that will be covered in this module.
Modul
e 2 Rational
Expressions
At the end of this module, the learners will:
1. illustrate rational expressions;
2. simplify rational expressions;
3. perform operations on rational algebraic expressions;
4. solve problems involving rational expressions.
Rational Expressions
Simplifying Rational expressions
Operation of Rational Expressions
Adding and Subtracting
Rational Expressions
Multiplying and Dividing
Rational Expressions

3. Module 2: Rational Expressions
Read and analyze each item carefully and encircle the letter of the
best answer.
Pre-test!

4. Ms. Lorie Jane L Letada

5. Module 2: Rational Expressions

6. Ms. Lorie Jane L Letada
A Rational Expressions is reduced to its simplest form if the
numerator and denominator have no common factors. Putting a rational
expression in its simplest form is important as this will make operations
easier.
Simplify the following in its simplest form.
1.
4𝑎
12𝑏
2.
9𝑚
21𝑛
3.
4𝑤2
6𝑤−8𝑤2
Lesson
1
Simplifying Rational
Expressions
Step 1
Factor thenumerator and
denominator and get theGCF.
4a = (2) (2) a
12b= (3) (2) (2) b
GCF: (2) (2)
GCF: 4
Thus, the common factor is 4.
Step 2:
Divideout thecommon factor.
4𝑎
12𝑏
=
4 ( 𝑎 )
4 (3𝑏)
=
𝑎
3𝑏
Thus, the simplify is
𝑎
3𝑏
.
Step 1
Factor thenumerator and
denominator and get the GCF.
9m =
21n =
GCF:
GCF:
Thus, the common factor is __.
Step 2:
Divide out the common factor.
Step 1
Get theGCF of each term.
4𝑤2
= (2) (2) (w) (w)
6w = (3) (2) (w)
8𝑤2
= (2)(2) (2)(w)
GCF: (2) (w)
GCF: 2w
Thus, the common factor is 2w.
Step 2:
Factor the common
numerator and
denominator.
4𝑤2
6𝑤−8𝑤2 =
2𝑤 ( 2)
2𝑤 (3−4𝑤 )
Step 3:
Divide out the common
factor.
4𝑤2
6𝑤−8𝑤2 =
2𝑤 ( 2)
2𝑤 (3−4𝑤 )
=
2
3−4𝑤
Thus, the simplify is
2
3−4𝑤
.

7. Module 2: Rational Expressions
4.
6𝑝2
2𝑝−4𝑝3
5.
2𝑥−2𝑦
𝑦2− 𝑥2
6.
3𝑥+ 3𝑦
𝑥2− 𝑦2
7.
4𝑝+ 4𝑞
𝑝2 − 𝑞2
Step 1
Get theGCF of each term.
6𝑝2
=
2p =
4𝑝3
=
GCF:
GCF:
Step 2:
Factor the common
numerator and
denominator.
6𝑝2
2𝑝−4𝑝3 =
Step 3:
Divide out the common factor.
Step 1
Factor thecommon numerator
and denominator.
2𝑥−2𝑦
𝑦2 − 𝑥2 =
2 (𝑦−𝑥)
( 𝑦−𝑥 )(𝑦+ 𝑥)
Step 2:
Divide out the common factor.
2𝑥−2𝑦
𝑦2 − 𝑥2 =
2 (𝑦−𝑥)
( 𝑦−𝑥 )(𝑦+ 𝑥)
=
2
(𝑥+ 𝑦)
You can only apply the
difference of two squares if:
*The two terms are both
perfect squares.
*The operation is subtraction.
Example:
𝑦2
− 𝑥2
= (y – x ) ( y + x )
Step 1
Factor thecommon numerator
and denominator.
3𝑥+ 3𝑦
𝑥2 − 𝑦2 =
Step 2:
Divide out the common factor. To factor 𝑥2
− 𝑦2
, get
the square root of each term.
First Term: √x2 = x
Second Term: √y2= y
Usingx and y, form the
sum (x + y) & thedifference
(x - y).
Thus, x2
- y2
= (x + y) (x – y).
Step 1
Factor thecommon numerator
and denominator.
4𝑝+ 4𝑞
𝑝2− 𝑞2 =
Step 2:
Divide out the common factor.
A variable
is perfect
square if
its
exponent
is even
number.

8. Ms. Lorie Jane L Letada
8.
𝑚2+𝑚−6
𝑚2−7𝑚+10
9.
𝑛2−7𝑛−30
𝑛2−5𝑛−24
10.
𝑏2−49
𝑏2−2𝑏−35
Simplify each rational expression.
1.
5𝑎𝑏2
20𝑎𝑏
= 5.
𝑥−1
5𝑥−5
=
2.
5𝑔ℎ
10𝑔
= 6.
𝑛2+2𝑛−15
𝑛2−𝑛−6
=
3.
15𝑎𝑐2
45𝑎𝑏3 𝑐
= 7.
𝑐2+7𝑐+10
𝑐2+5𝑐
=
4.
𝑥2−9
𝑥2−𝑥−6
= 8.
𝑞2+8𝑞+12
𝑞2+3𝑞−18
=
Skill Booster!
Step 1
Factor thecommon numeratorand
denominator.
𝑚2+𝑚−6
𝑚2−7𝑚+10
=
( 𝑚+3)(𝑚−2 )
( 𝑚−5)(𝑚−2)
Step 2:
Divide out the common factor.
𝑚2+𝑚−6
𝑚2−7𝑚+10
=
( 𝑚+3)(𝑚−2 )
( 𝑚−5)(𝑚−2)
=
(𝑚+3)
(𝑚−5)
Step 1
Factor thecommon numeratorand
denominator.
𝑛2−7𝑛−30
𝑛2−5𝑛−24
=
Step 2:
Divide out the common factor.
Step 1
Factor thecommon numeratorand
denominator.
𝑏2−49
𝑏2−2𝑏−35
=
(𝑏−7(𝑏+7)
( 𝑏−7)(𝑏+5 )
Step 2:
Divide out the common factor.
𝑏2−49
𝑏2−2𝑏−35
=
(𝑏−7 (𝑏+7)
( 𝑏−7 )(𝑏+5)
=
( 𝑏+7)
(𝑏+5)
To factor 𝑚2
+ 𝑚 − 6 ,
find any factors of -6
that when you add the
result is 1,
Hence, the correct pair
is 3 and -2.
So, the factors of
𝑚2
+ 𝑚 − 6 = (m +3)
(m -2).
* When you multiply
positive & negative
numbers, the result is
negative.
Example: (5 )(-3)=-15
*When you add positive
& negative numbers,the
sign used in the result
will be based which
absolute num. is greater.
Examples: 5 +-3 =2
*When you add positive
& negative numbers,the
sign used in the result
will be based which
absolute num. is greater.
Examples: 5 +-3 =2

9. Module 2: Rational Expressions
To multiply rational expressions, recall the rules for multiplying
fractions. If the denominators are not equal to zero, then we simply
multiply the numerators and denominators. The same rule applies to
rational expressions.
If a, b, c, and d represent polynomials where b ≠ 0 and d ≠ 0. Then,
Multiply each rational expression and simplify.
1.
𝟓
𝟏𝟐
∗
𝟖
𝟏𝟓
=
2.
𝟔
𝟏𝟎
∗
𝟓
𝟏𝟓
=
3.
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
Lesson
2
Multipication of Rational
Expressions
𝒂
𝒃
∗
𝒄
𝒅
=
𝒂𝒄
𝒃𝒅
Step 1 Factor each expression.
𝟓
𝟏𝟐
∗
𝟖
𝟏𝟓
=
𝟓
( 𝟒)(𝟑)
∗
( 𝟒) (𝟐)
( 𝟑) (𝟓)
Step 2 Cancel all common
factors.
𝟓
( 𝟒)(𝟑)
*
( 𝟒) (𝟐)
( 𝟑) (𝟓)
Step 3: Write what is left.
𝟓
𝟏𝟐
∗
𝟖
𝟏𝟓
=
𝟐
( 𝟑) (𝟑)
=
𝟐
𝟗
Step 1 Factor each expression. Step 2 Cancel all common
factors.
Step 3: Write what is left.
Step 1: Factor each expression.
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
( 𝟓 )( 𝟑)
( 𝟐)(𝒚 𝟐)
∗
( 𝒚 𝟐)(𝒚 𝟐
)
( 𝟓)( 𝟒)(𝒚 𝟐)
Step 2 Cancel all common factors.
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
( 𝟓 ) ( 𝟑)
( 𝟐) (𝒚 𝟐)
∗
( 𝒚 𝟐) (𝒚 𝟐
)
( 𝟓) ( 𝟒) (𝒚 𝟐)
Step 3: Write what is left.
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
𝟑
(𝟐)(𝟒)
=
𝟑
𝟖

10. Ms. Lorie Jane L Letada
4.
𝟏𝟐
𝟑𝒘 𝟑 ∗
𝟒𝒘 𝟒
𝟏𝟓𝒘 𝟐 =
𝟓.
𝟒𝒙 𝟐
−𝟗
𝒙 𝟐 −𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
𝟔.
𝒙+𝟓
𝟒
∗
𝟏𝟐𝒙 𝟐
𝒙 𝟐 +𝟕𝒙+𝟏𝟎
=
𝟕.
𝟑𝒔
𝟒𝒔+𝟏
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 =
𝟖 .
𝟑
𝒙−𝒚
∗
(𝒙−𝒚 ) 𝟐
𝟔
=
Step 1: Factor each
expression.
Step 2 : Cancel all common
factors.
Step 3: Write what is left.
Step 1: Factor each expression.
𝟒𝒙 𝟐
−𝟗
𝒙 𝟐 −𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
( 𝟐𝒙−𝟑 )(𝟐𝒙+𝟑)
( 𝒙−𝟑)(𝒙−𝟐)
*
𝟑 ( 𝒙−𝟐 )
𝟒(𝟐𝒙+𝟑)
Step 2 : Cancel all common factors.
𝟒𝒙 𝟐
−𝟗
𝒙 𝟐 −𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
( 𝟐𝒙−𝟑 )(𝟐𝒙+𝟑)
( 𝒙−𝟑)(𝒙−𝟐)
*
𝟑 ( 𝒙−𝟐 )
𝟒(𝟐𝒙+𝟑)
Step 3: Write what is left.
𝟒𝒙 𝟐
−𝟗
𝒙 𝟐−𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
( 𝟐𝒙−𝟑 )
( 𝒙−𝟑)
∗
𝟑
𝟒
=
𝟑 ( 𝟐𝒙−𝟑 )
𝟒 ( 𝒙−𝟑 )
Showyour solution here .….
Step 1: Factor each expression.
𝟑𝒔
𝟒𝒔+𝟐
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 = 𝟑𝒔
𝟐(𝟐𝒔+𝟏)
∗
(𝟐𝒔+𝟏)
𝒔 ( 𝟑𝒔 )
Step 2 : Cancel all common factors.
𝟑𝒔
𝟒𝒔+𝟐
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 = 𝟑𝒔
𝟐(𝟐𝒔+𝟏)
∗
(𝟐𝒔+𝟏)
𝒔 ( 𝟑𝒔 )
𝟑𝒔
𝟒𝒔+𝟐
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 =
𝟏
𝟐 ( 𝒔)
=
𝟏
𝟐𝒔
Showyour solution here .….
Cancelation involving
rationalexpressions should
be applied carefully. In the
expression
2𝑥−3
𝑥−3
,youcannot
cancel x in the numerator
and denominator even
though
𝑥
𝑥
is equal to1
because it is not a factor of
2𝑥−3
𝑥−3
. 𝐼𝑛 𝑓𝑎𝑐𝑡,
2𝑥−3
𝑥−3
is already in simplest
form.
Factor, in
mathematics, a
number or
algebraic
expression that
divides another
number or
expression
evenly—i.e., with
no remainder. For
example, 3 and 6
are factors of 12
because 12 ÷3 =4
exactly and 12 ÷6
= 2 exactly.

11. Module 2: Rational Expressions
Math Focus
To multiply rational expressions:
 Write each numerator and denominator in factored form.
 Divide out any numerator factor with any matching
denominator factor.
 Multiply numerator by numerator and denominator by
denominator.
 Simplify as needed.
Multiply each rational expression and simplify .
1.
𝟏𝟒
𝟐𝟕
∗
𝟑
𝟕
= 5.
(𝒓 𝟐
+𝟑𝒓+𝟐 )
𝒓−𝟏
*
𝒓+𝟑
𝒓 𝟐+𝟓𝒓+𝟔
=
2.
𝟏𝟐𝒏
𝟒𝒎 𝟐 ∗
𝟖𝒎 𝟒
𝟏𝟓𝒏 𝟐 = 6.
(𝒂 𝟐
−𝟏 )
𝟏𝟔 𝒂
*
(𝟒𝒂 𝟐
𝟕𝒂+𝟕
=
3.
𝟏𝟐𝒏
𝟒𝒎 𝟐 ∗
𝟖𝒎 𝟒
𝟏𝟓𝒏 𝟐 = 7.
(𝒚 𝟐
+𝟒𝒚+𝟒 )
𝒚+𝟑
*
𝟒𝒚+𝟏𝟐
𝒚+𝟐
= =
4.
𝟐𝒎+𝟓
𝟑𝒎−𝟔
*
(𝒎 𝟐
+𝒎−𝟔 )
𝟒
=
Skill Booster!

12. Ms. Lorie Jane L Letada
In dividing rational expressions, observe the same rules as when
dividing fractions.
If the denominators are not equal to zero, then
Lesson
3
Division of Rational
Expressions
𝒂
𝒃
÷
𝒄
𝒅
=
𝒂
𝒃
∗
𝒅
𝒄
=
𝒂𝒅
𝒃𝒄