Directions: Write the Rational Algebraic Expressions in simplest form.
Given Solution Scratch
1.)
24𝑚2𝑝2
36𝑚𝑝2 =
𝟏𝟐𝒎𝒑𝟐(2𝑚)
𝟏𝟐𝒎𝒑𝟐(3)
Numerator 24𝑚2
𝑝3 = (2) (2) (2) (3) (m) (m) (p) (p)
Denominator 36𝑚 𝑝3 = (2) (2) (3) (3) (m) (p) (P)
GCMF = (2) (2) (3) (m) (p) (p)
= 𝟏𝟐𝒎𝒑𝟐
=
𝟏𝟐𝒎𝒑𝟐(2𝑚)
𝟏𝟐𝒎𝒑𝟐(3)
Simplified Form
=
2𝑚
3
2.)
4𝑏𝑝2+6𝑏2𝑝
6𝑝2+2𝑝
=
𝟐𝒃𝒑(𝟐𝒑 +𝟑𝒃)
𝟐𝒑(𝟑𝒑 +𝟏)
Numerator 4𝑏𝑝2 = (2) (2) b (p) (p)
6𝑏2
𝑝 = (2) (3) (b) (b) (p)
GCMF = (2) (b) (p)
= 𝟐𝒃𝒑
Factored
Form of
the
Numerator
= 𝟐𝒃𝒑(2p +3b)
=
𝟐𝒃𝒑(𝟐𝒑 +𝟑𝒃)
𝟐𝒑(𝟑𝒑 +𝟏)
Simplified Form
=
𝒃(𝟐𝒑 +𝟑𝒃)
(𝟑𝒑 +𝟏)
Denominator 6𝑝2 = (2) (3) (p) (p)
2𝑝 = (2) (p)
GCMF = (2) (p)
= 𝟐𝒑
Factored
Form of the
Numerator
= 𝟐𝒑(3p +1)

Given Solution Scratch
3.)
4𝑟2− 9𝑐2
8𝑟3+ 27𝑐3
=
(𝟐𝒓+𝟑𝒄)(2𝑟−3𝑐)
(2𝑟+3𝑐 )( 4𝑟2 – 6𝑟𝑐+9𝑐2)
Numerator Difference
of Two
Squares
Formula
𝑎2
− 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
4𝑟2
− 9𝑐2 = (2𝑟 + 3𝑐)(2𝑟 − 3𝑐)
(2𝑟)(2𝑟) (3𝑐)(3𝑐)
𝒂 𝒃
Denominator Sum of
Two
Cubes
8𝑟3
+ 27𝑐3 = (2𝑟)3
+ (3𝑐)3
(2r)(2𝑟)(2𝑟) (3𝑐)(3𝑐)(3𝑐)
𝒇𝒊𝒓𝒔𝒕 𝒃𝒂𝒔𝒆 𝒔𝒆𝒄𝒐𝒏𝒅 𝒃𝒂𝒔𝒆
= (2𝑟 3𝑐 )( )
= (2𝑟 3𝑐 )( 2𝑟.2𝑟 )
= (2𝑟 3𝑐 )( 4𝑟2
)
= (2𝑟 3𝑐 )( 4𝑟2
2𝑟. 3𝑐 )
= (2𝑟 3𝑐 )( 4𝑟2
6𝑟𝑐 )
= (2𝑟 3𝑐 )( 4𝑟2
6𝑟𝑐 3𝑐. 3𝑐) )
= (2𝑟 3𝑐 )( 4𝑟2
6𝑟𝑐 9𝑐2
)
= (2𝑟 + 3𝑐 )( 4𝑟2
– 6𝑟𝑐 + 9𝑐2
)
S O AP
=
(𝟐𝒓+𝟑𝒄)(2𝑟−3𝑐)
(2𝑟+3𝑐 )( 4𝑟2 – 6𝑟𝑐+9𝑐2)
Simplified
Form
=
2𝑟−3𝑐
4𝑟2 – 6𝑟𝑐+9𝑐2

Given Solution Scratch
4.)
4𝑑2+2𝑑− 6
2𝑑2+5𝑑+ 3
=
(2𝑑+3)(2𝑑−2)
(2𝑑+3)(𝑑+1)
Numerator Trinomial (2𝑑 + 3)
(2𝑑 − 2)
Factors of
the
Trinomial
(2𝑑 + 3)(2𝑑 − 2)
Denominator Trinomial (2𝑑 + 3)
(𝑑 + 1)
Factors of
the
Trinomial
(2𝑑 + 3)(𝑑 + 1)
=
(2𝑑+3)(2𝑑−2)
(2𝑑+3)(𝑑+1)
Simplified
Form
=
2𝑑−2
𝑑+1

Given Solution Scratch
5.)
2𝑑− 6
6 − 2𝑑
=
(2𝑑−6)
−1(2𝑑−6)
Retain Numerator 2𝑑 − 6
Change the Denominator 6 − 2𝑑
a. 2𝑑 6
b. 2𝑑 − 6
c. −1(2𝑑 − 6)
=
(2𝑑−6)
−1(2𝑑−6)
Simplified
Form
=
1
−1
= −1