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Grade 8 – Mathematics
Quarter I
SIMPLIFYING RATIONAL
ALGEBRAIC EXPRESSIONS
1. illustrate rational algebraic
expressions; and
2. simplify rational algebraic
expressions.
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,
DOMAIN
- is the set of all possible values of the
variable that are allowed.
- restricted values can make a rational
expression undefined.
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.
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
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
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
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
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)
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
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

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Simplifying Rational Algebraic 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