The document outlines topics related to simplifying rational algebraic expressions, targeting eighth-grade mathematics. It explains the definition of rational expressions, exemplifies how to identify and restrict values that make expressions undefined, and provides steps for simplifying these expressions through factoring and canceling common factors. Multiple examples are given to illustrate the simplification process for various polynomial expressions.

1. Grade 8 – Mathematics
Quarter I
SIMPLIFYING RATIONAL
ALGEBRAIC EXPRESSIONS

2. 1. illustrate rational algebraic
expressions; and
2. simplify rational algebraic
expressions.

3. RATIONAL ALGEBRAIC EXPRESSIONS
- A rational expression in one variable is
an expression that can be written in the
form
𝑷
𝑸
, where P and Q are polynomials.
𝟒
𝒙 −𝟐
𝒙 𝟐
+ 𝟐𝒙 − 𝟑𝟓
𝟑𝒙 + 𝟒
𝟒𝒙
𝒙 𝟐 + 𝟗
not UNDEFINEDRemember, Q ≠ 0,

4. DOMAIN
- is the set of all possible values of the
variable that are allowed.
- restricted values can make a rational
expression undefined.

5. UNDEFINED
To make a rational expression
𝒙 𝟐−𝟐
𝒙 −𝟒
UNDEFINED,
x – 4 = 0equate the denominator to zero
solve for the variable x = 0 + 4
x = 4
Domain: All real numbers except 4.

6. UNDEFINED
To make a rational expression
𝟐𝒙
(𝒙 𝟐+𝟓𝒙+𝟔)
UNDEFINED
𝒙 𝟐
+ 𝟓𝒙 + 𝟔equate the denominator to zero
solve for the variable
Domain: All real numbers except -2 and -3.
(x + 2)(x + 3) = 0
x + 2 = 0 x + 3 = 0
x = -2 x = -3

7. UNDEFINED
To make a rational expression
𝟐𝒙
(𝒙 𝟐−𝟗)
UNDEFINED,
𝒙 𝟐
− 9 = 0equate the denominator to zero
solve for the variable
x + 9 = 0
Domain: All real numbers except -9 and 9.
(x + 9)(x -9) = 0
x - 9 = 0
x = - 9 x = 9

8. SIMPLIFYING RATIONAL ALGEBRAIC
EXPRESSIONS
Factor the numerator
and denominator
Example:
𝟔𝒙+𝟏𝟐
𝟑
=
𝟑(𝟐𝒙+𝟔)
𝟑
Cancel out or divide
the common factor. =
𝟑(𝟐𝒙+𝟔)
𝟑
Simplify. = 2x + 6
GCF: 3

9. SIMPLIFYING RATIONAL ALGEBRAIC
EXPRESSIONS
Factor the numerator
and denominator
Example:
𝒙 𝟐−𝟔𝒙+𝟖
𝟒𝒙−𝟖
=
(𝒙−𝟒)(𝒙−𝟐)
𝟒(𝒙−𝟐)
Cancel out or divide
the common factor.
=
(𝒙−𝟒)(𝒙−𝟐)
𝟒(𝒙−𝟐)
Simplify. =
(𝒙−𝟒)
𝟒
GCF: 4

10. SIMPLIFYING RATIONAL ALGEBRAIC
EXPRESSIONS
Factor the numerator
and denominator
Example:
𝒂 𝟐+𝒂−𝟔
𝒂 𝟐−𝟐𝒂−𝟏𝟓
=
(𝒂+𝟑)(𝒂−𝟐)
(𝒂−𝟓)(𝒂+𝟑)
Cancel out or divide
the common factor.
=
(𝒂+𝟑)(𝒂−𝟐)
(𝒂−𝟓)(𝒂+𝟑)
Simplify. =
(𝒂−𝟐)
(𝒂−𝟓)

11. SIMPLIFYING RATIONAL ALGEBRAIC
EXPRESSIONS
Factor the numerator
and denominator
Example:
𝟑𝒙 𝟐 𝒚 𝟒
𝟗𝒙 𝟑 𝒚 𝟐
=
𝟑·𝒙 𝟐·𝒚 𝟐·𝒚 𝟐
𝟗·𝒙 𝟐·𝒙·𝒚 𝟐
Cancel out or divide
the common factor.
=
𝟑·𝒙 𝟐·𝒚 𝟐·𝒚 𝟐
𝟗·𝒙 𝟐·𝒙·𝒚 𝟐
Simplify. =
𝒚 𝟐
𝟑𝒙

12. SIMPLIFYING RATIONAL ALGEBRAIC
EXPRESSIONS
Factor the numerator
and denominator
Example:
𝟏𝟐𝒙 𝟓 𝒚 𝟒
𝟐𝒙 𝟑 𝒚 𝟕
=
𝟏𝟐·𝒙 𝟑·𝒙 𝟐·𝒚 𝟒
𝟐·𝒙 𝟑·𝒚 𝟒·𝒚 𝟑
Cancel out or divide
the common factor.
=
𝟏𝟐·𝒙 𝟑·𝒙 𝟐·𝒚 𝟒
𝟐·𝒙 𝟑·𝒚 𝟒·𝒚 𝟑
Simplify. =
𝟔𝒙 𝟐
𝒚 𝟑