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The problems and solutions in Algebraic and Rational Expressions
More than 80 problems

 Before this book you have to read :
“Operations with Algebraic and Rational Expressions”

AUGUST 26, 2022
HELMA ISAZEI
Helma_isazei@outlook
...
2
Addition and subtraction of monomials:
...
4
Multiply polynomials by monomials
...
5
DIVIDING ALGEBRAIC EXPRESSIONS
...
7
Dividing polynomials by integers and monomials
...
10
Simplifying rational expressions using algebraic Identities
...
11
Multiplying rational expressions
...
13
Sum and Difference of Rational Expressions
...
14
To add or subtract two rational expressions with unlike denominator
...
17

1|Page

Operations with Algebraic and Rational Expressions

HELMA NOTE

ADDING AND SUBTRACTING ALGEBRAIC EXPRESSIONS
Addition and subtraction of monomials:
Example 1:

𝑚 + 2 + 3𝑚 − 7 = 4𝑚 − 5

Example 2:

3𝑎 + 2𝑏 + (6𝑎 − 6𝑏) = 3𝑎 + 2𝑏 + 6𝑎 − 6𝑏 = 9𝑎 − 4𝑏

Example 3:

2𝒙2 − 5𝒙 − 9 − (𝒙2 − 2𝒙 − 1) =
2𝒙2 − 5𝒙 − 9 − 𝒙2 + 2𝒙 + 1 =
𝒙2 − 3𝒙 − 8
Example 4:

𝒂𝟐   −   𝟑𝒂𝒙 + 𝒙𝟐   − [𝟒𝒂𝟐 + 𝟓𝒂𝒙 − (𝟑𝒂𝟐 − 𝟖𝒂𝒙)] =

𝒂𝟐   −   𝟑𝒂𝒙 + 𝒙𝟐   − [𝟒𝒂𝟐 + 𝟓𝒂𝒙 − 𝟑𝒂𝟐 + 𝟖𝒂𝒙] =
𝒂𝟐   −   𝟑𝒂𝒙 + 𝒙𝟐   − 𝟒𝒂𝟐 − 𝟓𝒂𝒙 + 𝟑𝒂𝟐 − 𝟖𝒂𝒙 = 𝒙𝟐 −   𝟏𝟔𝒂𝒙  

Example 5:

  − [𝟒𝒃𝟐 + 𝟓𝒂 − 𝟒 − (𝒃𝟐 − 𝟑𝒂 + 𝟔)] =
  − [𝟒𝒃𝟐 + 𝟓𝒂 − 𝟒 − 𝒃𝟐 + 𝟑𝒂 − 𝟔] =
  − 𝟒𝒃𝟐 − 𝟓𝒂 + 𝟒 + 𝒃𝟐 − 𝟑𝒂 + 𝟔 =
  − 𝟑𝒃𝟐 − 𝟖𝒂 + 𝟏𝟎

2|Page

HELMA NOTE

Operations with Algebraic and Rational Expressions

Example 6:

3𝒙2 − 4𝒙 + 8 − (5𝒙2 − 6𝒙 − 1) =
3𝒙2 − 4𝒙 + 8 − 5𝒙2 + 6𝒙 + 1 =
−2𝒙2 + 2𝒙 + 9

Example 7:

5𝒙2 + 10𝒙𝒚 + 25 − (𝒙2 + 12𝒙𝒚 + 15) =
5𝒙2 + 10𝒙𝒚 + 25 − 𝒙2 − 12𝒙𝒚 − 15
= 4𝒙2 − 2𝒙𝒚 + 10

Example 8:

(𝒚𝟐 + 𝒚 + 𝟏) + (𝒚𝟐 − 𝒚 − 𝟏) =
𝒚𝟐 + 𝒚 + 𝟏 + 𝒚𝟐 − 𝒚 − 𝟏 = 𝟐𝒚𝟐

Example 9:

(𝟕𝒚𝟐 + 𝟏𝟎𝒚 − 𝟓) − (𝟓𝒚𝟐 − 𝟖𝒚 − 𝟏)
𝟕𝒓𝟐 + 𝟏𝟎𝒓 − 𝟓 − 𝟓𝒓𝟐 + 𝟖𝒚 + 𝟏
= 𝟐𝒓𝟐 + 𝟏𝟖𝒚 − 𝟒



Example 10:

(𝟐𝒙 + 𝟓𝒙𝒚 − 𝟑𝒚) + (𝟏𝟐𝒙 − 𝟐𝒙𝒚 − 𝟓𝒚) =
𝟐𝒙 + 𝟓𝒙𝒚 − 𝟑𝒚 + 𝟏𝟐𝒙 − 𝟐𝒙𝒚 − 𝟓𝒚 =
𝟏𝟒𝒙 + 𝟑𝒙𝒚 – 𝟖𝒚

3|Page

HELMA NOTE

Operations with Algebraic and Rational Expressions

MULTYPLY ALGEBRAIC EXPRESSIONS
Multiply polynomials by monomials
Examples 11:

3𝑦(𝑦 2 − 3𝑦 + 1) =
3𝑦 3 − 9𝑦 2 + 3𝑦
Examples 12:

4𝑥(2𝑥 − 3) − 3𝑥(𝑥 + 2) =
𝟖𝒙𝟐 − 𝟏𝟐𝒙 − 𝟑𝒙𝟐 − 𝟔𝒙 = 𝟓𝒙𝟐 − 𝟏𝟖𝒙

Examples 13:

𝟒𝒓 𝟐 + 𝟑 − 𝟓𝒓(𝒓 − 𝟖) =
𝟒𝒓 𝟐 + 𝟑 − 𝟓𝒓 𝟐 + 𝟒𝟎𝒓 =
−𝒓 𝟐 + 𝟒𝟎𝒓 + 𝟑

Examples 14:

𝟐𝒙𝟐 (𝒙 − 𝟓) − 𝒙(𝟑𝒙𝟐 − 𝟐) =
𝟐𝒙𝟑 − 𝟏𝟎𝒙𝟐 − 𝟑𝒙𝟑 + 𝟐𝒙 =
−𝒙𝟑 − 𝟏𝟎𝒙𝟐 + 𝟐𝒙

Examples 15:

𝟑𝒎(𝒎 + 𝒂 − 𝟓) =
𝟑𝒎𝟐 + 𝟑𝒎𝒂 − 𝟏𝟓𝒎

Examples 16:

𝟏
𝟖𝒚𝟐 𝟔𝒚 𝟒
𝟐
𝟐
(𝟖𝒚 + 𝟔𝒚 − 𝟒) =
+
−
= 𝟒𝒚 + 𝟑 −
𝟐𝒚
𝟐𝒚 𝟐𝒚 𝟐𝒚
𝒚
4|Page

Operations with Algebraic and Rational Expressions

HELMA NOTE
Examples 17:

𝒙𝟐 𝒚(𝒙𝟐 − 𝟐𝒙𝒚 + 𝒚𝟐 ) − 𝒙𝒚𝟐 (𝟐𝒙𝟐 − 𝟑𝒙𝒚 − 𝒚𝟐 ) =
𝒙𝟒 𝒚 − 𝟐𝒙𝟑 𝒚 + 𝒙𝟐 𝒚𝟑 − 𝟐𝒙𝟑 𝒚𝟐 + 𝟑𝒙𝟐 𝒚𝟑 + 𝒙𝒚𝟐 =
𝒙𝟒 𝒚 − 𝟐𝒙𝟑 𝒚 + 𝟒𝒙𝟐 𝒚𝟑 − 𝟐𝒙𝟑 𝒚𝟐 + 𝒙𝒚𝟐 Done

Multiply polynomials by polynomials
Examples 18:

(𝒙 + 𝟐 )(𝒚 + 𝟑 ) = 𝒙𝒚 + 𝟑𝒙 + 𝟐𝒚 + 𝟔
Examples 19:

(𝟐𝒙 + 𝟏 )(𝒚 + 𝟑 ) = 𝟐𝒙𝒚 + 𝟔𝒙 + 𝒚 + 𝟑

Examples 20:

(𝒚 + 𝟏 )(𝒚 + 𝟑 ) = 𝒚𝟐 + (𝟏 + 𝟑)𝒚 + (𝟏 × 𝟑) = 𝒚𝟐 + 𝟒𝒚 + 𝟑

(Identity IV)

Examples 21:

(𝒙 + 𝒂 )(𝒙 + 𝒃 ) = 𝒙𝟐 + 𝒃𝒙 + 𝒂𝒙 + 𝒂𝒃
= 𝒙𝟐 + 𝒙(𝒃 + 𝒂) + 𝒂𝒃

(Identity IV)

Examples 22:
𝟐

𝟐

(𝒙 + √𝟓 ) = 𝒙𝟐 + 𝟐(𝒙)(√𝟓) + (√𝟓) = 𝒙𝟐 − 𝟐√𝟓𝒙 + 𝟓 (Identity I)

Examples 23:

(𝒙 + 𝟓 )(𝒙 − 𝟓 ) = 𝒙𝟐 − 𝟓𝟐
= 𝒙𝟐 − 𝟐𝟓
5|Page

(Identity III)

HELMA NOTE

Operations with Algebraic and Rational Expressions

Examples 24:

(𝒙 − 𝟓 )𝟐 = 𝒙𝟐 − 𝟐(𝒙)(𝟓) + 𝟓𝟐
= 𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓

(Identity II)

Examples 25:

𝟏 𝟐
𝟏
𝟏 𝟐
𝟐
𝟐
𝟐
(𝒙 + ) = (𝒙 ) + 𝟐(𝒙 ) ( ) + ( )
𝟐
𝟐
𝟐
𝟐

= 𝒙𝟒 + 𝒙𝟐 +

𝟏
𝟒

(Identity I)

Examples 26:

𝟐 𝟐
𝟐
𝟐 𝟐
𝟐
(𝒚 − ) = (𝒚) − 𝟐(𝒚) ( ) + ( )
𝟑
𝟑
𝟑
𝟐

𝟒

𝟑

𝟗

= 𝒚𝟐 − 𝒚 +

(Identity II)

Examples 27:

(𝒙 + 𝟐 )(𝟐𝒙 + 𝟑𝒙 + 𝟓 ) = 𝟐𝒙𝟐 + 𝟑𝒙𝟐 + 𝟓𝒙 + 𝟒𝒙 + 𝟔𝒙 + 𝟏𝟎
= 𝟓𝒙𝟐 + 𝟏𝟓𝒙 + 𝟏𝟎
Examples 28:

(𝟑𝒙𝟐 + 𝟐𝒙 + 𝟓 )(𝒙𝟐 + 𝟑𝒙 + 𝟐 )
= 𝟑𝒙𝟒 + 𝟗𝒙𝟑 + 𝟔𝒙𝟐 + 𝟐𝒙𝟑 + 𝟔𝒙𝟐 + 𝟒𝒙 + 𝟓𝒙𝟐 + 𝟏𝟓𝒙 + 𝟏𝟎
= 𝟑𝒙𝟒 + 𝟏𝟏𝒙𝟑 + 𝟏𝟕𝒙𝟐 + 𝟏𝟗𝒙 + 𝟏𝟎

Examples 29:

(𝐤 – 𝟐)(𝟖𝐤 – 𝟕) = 𝟖𝒌𝟐 − 𝟕𝒌 − 𝟏𝟔𝒌 + 𝟏𝟒 = 𝟖𝒌𝟐 − 𝟐𝟑𝒌 + 𝟏𝟒

6|Page

HELMA NOTE

Operations with Algebraic and Rational Expressions

DIVIDING ALGEBRAIC EXPRESSIONS
Dividing monomials by monomials
Examples 30:

𝟏𝟖𝒚𝟓 𝟑𝒚𝟐
𝟏𝟖𝒚 ÷ 𝟔𝟎𝒚 =
=
𝟔𝟎𝒚𝟑
𝟏𝟎
𝟓

𝟑

Examples 31:

𝟔𝒙𝟕
𝟔𝒙 ÷ 𝟑𝒙 = 𝟒 = 𝟐𝒙𝟑
𝟑𝒙
𝟕

𝟒

Examples 32:

𝟐𝟎𝒎𝟕 𝟒𝒎𝟓
𝟐𝟎𝒎 ÷ 𝟐𝟓𝒎 =
=
𝟐𝟓𝒎𝟐
𝟓
𝟓

𝟑

Examples 33:

𝟖𝒙𝟓 𝒚𝟏𝟐
= −𝟒𝒙𝟐 𝒚𝟐
𝟑
𝟏𝟎
−𝟐𝒙 𝒚
Examples 34:

𝟖𝒙𝟒 𝒚𝟒 𝒛
= 𝟐𝒙𝒛−𝟐
𝟑
𝟒
𝟑
𝟒𝒙 𝒚 𝒛

Examples 35:

𝟔𝒙𝟐 𝒚𝟑
𝟏 −𝟑
=
𝒙 𝒚
𝟏𝟐𝒙𝟓 𝒚𝟐 𝟐

7|Page

𝒙−𝟑 =

𝟏
𝒙𝟑

HELMA NOTE

Operations with Algebraic and Rational Expressions

Examples 36:

𝟔𝒂𝟒 𝒃𝟐 𝟔𝒂𝟑
=
𝟓𝒂𝒃𝟐
𝟓
Examples 37:

𝒎𝟕 𝒏𝟓
= 𝒎𝟔 𝒏𝟒
𝒎𝒏

Dividing polynomials by integers and monomials
Examples 38:

(𝟐𝟎𝒙𝒚 + 𝟏𝟔𝒙) ÷ 𝟒𝒙 =

𝟐𝟎𝒙𝒚 𝟏𝟔𝒙
+
= 𝟓𝒚 + 𝟒
𝟒𝒙
𝟒𝒙

Examples 39:

𝟓𝒙𝒚𝟒 + 𝟐𝟓𝒙𝟐 𝒚𝟓 + 𝟒𝟎𝒙𝟑 𝒚𝟐
= 𝒚𝟑 + 𝟓𝒙𝒚𝟒 + 𝟖𝒙𝟐 𝒚
𝟓𝒙𝒚

Examples 40:

𝟏𝟖𝒚𝟕 𝟗𝒚𝟑
𝒚𝟓
− 𝟑+ 𝟑
(𝟏𝟖𝒚 − 𝟗𝒚 + 𝒚 ) ÷ 𝟗𝒚 =
𝟗𝒚𝟑
𝟗𝒚
𝟗𝒚
𝟕

𝟑

𝟓

𝟑

𝒚𝟐
= 𝟐𝒚 − 𝟏 +
𝟗
𝟒

Examples 41:

(𝟐𝒙 + 𝟒𝒚) ÷ 𝟐 =

8|Page

𝟐𝒙 𝟒𝒚
+
= 𝒙 + 𝟐𝒚
𝟐
𝟐

Operations with Algebraic and Rational Expressions

HELMA NOTE
Examples 42:

(𝟒𝟗𝒂𝟐

𝟐)

𝒃 − 𝟏𝟒𝒂𝒃 + 𝒂𝒃

𝟒𝟗𝒂𝟐 𝒃 𝟏𝟒𝒂𝒃 𝒂𝒃𝟐
÷ 𝟕𝒂𝒃 =
−
+
𝟕𝒂𝒃
𝟕𝒂𝒃 𝟕𝒂𝒃
= 𝟕𝒂 − 𝟐 +

𝒃
𝟕

Examples 43:

𝒙𝟑

(

+

𝟑

𝟓𝒙𝟐
𝟐

+

𝟕𝒙

𝒙𝟑

𝟑

𝟑(𝟐𝒙)

) ÷ 𝟐𝒙 =

+

𝟓𝒙𝟐
𝟐(𝟐𝒙)

+

𝟕𝒙
𝟑(𝟐𝒙)

=

𝒙𝟐
𝟔

+

Examples 44:

(𝟓𝟒𝒏𝟒 – 𝟑𝟔𝒏𝟐 – 𝟒𝟓𝒏) 𝟓𝟒𝒏𝟒 𝟑𝟔𝒏𝟐 𝟒𝟓𝒏
𝟓
𝟐
=
−
−
=
𝟔𝒏
−
𝟒
−
𝟗𝒏𝟐
𝟗𝒏𝟐
𝟗𝒏𝟐
𝟗𝒏𝟐
𝒏
Examples 45:

(𝒙𝟑

𝒙𝟑 𝒙𝟐
𝒙
𝒙𝟐 𝒙 𝟏
+ 𝒙 + 𝒙) ÷ 𝟐𝒙 =
+
+
=
+ +
𝟐𝒙 𝟐𝒙 𝟐𝒙
𝟐 𝟐 𝟐
𝟐

Examples 46:

𝟏𝟐𝒎𝒏 + 𝟐𝒎𝟐 𝒏𝟑 + 𝟏𝟒𝒎𝟐 𝒏𝟐
= 𝟔 + 𝒎𝒏𝟐 + 𝟕𝒎𝒏
𝟐𝒎𝒏
Examples 47:

(𝟗𝒛𝟒

𝟗𝒛𝟒 𝟓𝒛𝟐
𝟏
𝟓 𝟏
+ 𝟓𝒛 + 𝟏) ÷ 𝒛 = 𝟑 + 𝟑 + 𝟑 = 𝟗𝒛 + + 𝟑
𝒛
𝒛
𝒛
𝒛 𝒛
𝟐

𝟑

Examples 48:
𝟑

𝟐

(√𝟏𝟒𝒒 + √𝟏𝟖𝒒 + √𝟑𝟐𝒒) ÷ √𝟐𝒒 =

√𝟏𝟒𝒒𝟑

√𝟐𝒒
𝟑𝒒
√𝟗𝒒 √𝟏𝟔
= √𝟕𝒒𝟐 +
+
= √𝟕𝒒𝟐 +
+𝟐
𝟐
𝟐
𝟐

9|Page

√𝟐𝒒

+

√𝟏𝟖𝒒𝟐

+

√𝟑𝟐𝒒
√𝟐𝒒

𝟓𝒙
𝟒

+

𝟕
𝟔

HELMA NOTE

Operations with Algebraic and Rational Expressions

Rational Expressions
Simplifying rational expressions using algebraic Identities
Examples 49:

Identity I

(𝒙 + 𝟑)𝟐
(𝒙 + 𝟑)
𝒙𝟐 + 𝟔𝒙 + 𝟗
=
=
𝒙𝟐 + 𝟒𝒙 + 𝟑 (𝒙 + 𝟏)(𝒙 + 𝟑) (𝒙 + 𝟏)
Identity IV
Examples 50:

Identity III

𝒎𝟐 − 𝟏𝟔 (𝒎 − 𝟒)(𝒎 + 𝟒)
=
=𝒎+𝟒
𝒎−𝟒
𝒎−𝟒
Examples 51:

𝒚+𝟏
𝒚+𝟏
𝟏
=
=
(𝒚 − 𝟏)(𝒚 + 𝟏) 𝒚 − 𝟏
𝒚𝟐 − 𝟏

Examples 52:

(𝟐𝒙 − 𝟓)(𝟐𝒙 + 𝟓) (𝟐𝒙 + 𝟓)
𝟒𝒙𝟐 − 𝟐𝟓
=
=
𝟒𝒙𝟐 − 𝟒𝒙 − 𝟏𝟓 (𝟐𝒙 − 𝟓)(𝟐𝒙 + 𝟑) (𝟐𝒙 + 𝟑)

Examples 53:

𝒙𝟐 + 𝟒𝒙 + 𝟒 (𝒙 + 𝟑)(𝒙 + 𝟏) (𝒙 + 𝟑)
=
=
𝒙𝟐 − 𝟓𝒙 − 𝟔 (𝒙 + 𝟏)(𝒙 − 𝟔) (𝒙 − 𝟔)

Examples 54:

𝒙 − √𝟓
𝒙 − √𝟓
𝟏
=
=
𝒙𝟐 − 𝟓 (𝒙 − √𝟓)(𝒙 + √𝟓) (𝒙 + √𝟓)
10 | P a g e

HELMA NOTE

Operations with Algebraic and Rational Expressions

Simplifying an rational expressions using factorization
Examples 55:

𝟑𝒚 + 𝟗 𝟑(𝒚 + 𝟑)
=
=𝟑
𝒚+𝟑
𝒚+𝟑

Examples 56:

𝟏𝟖𝒙𝟐 + 𝟔𝒙 𝟔𝒙(𝟑𝒙 + 𝟏)
=
= 𝟑𝒙 + 𝟏
𝟔𝒙
𝟔𝒙
Examples 57:

𝒃−𝟓
𝒃−𝟓
𝒃−𝟓
=
=
= −𝟏
𝟓 − 𝒃 −(−𝟓 + 𝒃) −(𝒃 − 𝟓)

Examples 58:

𝟐𝒎𝟑 + 𝟐𝒎𝟐 − 𝟒𝒎 𝟐𝒎(𝒎𝟐 + 𝒎 − 𝟐) 𝟐𝒎((𝒎 − 𝟏)(𝒎 + 𝟐))
=
=
= 𝟐(𝒎 − 𝟏)
𝒎(𝒎 + 𝟐)
𝒎(𝒎 + 𝟐)
𝒎(𝒎 + 𝟐)

Multiplying rational expressions
Examples 59:

𝟓𝒙𝒚𝟑 𝟏𝟔𝒛𝟑 𝟐𝒚𝒛
×
=
𝟖𝒙𝟐 𝒛𝟐 𝟏𝟓𝒚𝟐
𝒙𝟑

Examples 60:

𝟑𝒂
𝒚
𝟑𝒂
𝒚
𝒂
×
=
×
=
𝟖𝒚𝟐 𝟏𝟐 𝟖𝒚𝟐 𝟏𝟐 𝟑𝟐𝒚
4

Examples 61:

𝟒
𝒃
𝟒𝒃
×
=
𝟑𝒃 − 𝟏 𝒃 − 𝟏 (𝟑𝒃 − 𝟏)(𝒃 − 𝟏)

11 | P a g e

HELMA NOTE

Operations with Algebraic and Rational Expressions

Examples 62:

𝟑𝒏𝟐 − 𝒏
𝒎
𝒏(𝟑𝒏 − 𝟏)
𝒎
𝒏
×
=
×
=
𝒎𝟐
𝟑𝒏 − 𝟏
𝒎𝟐
𝟑𝒏 − 𝟏 𝒎
Examples 63:

𝒙+𝟑
𝒙𝟐
𝒙+𝟑
𝒙𝟐
𝒙
× 𝟐
=
×
=
(𝒙 + 𝟑)(𝒙 − 𝟓) (𝒙 − 𝟓)
𝒙
𝒙 − 𝟐𝒙 − 𝟏𝟓
𝒙
Identity IV
Examples 64:

𝒙−𝟐
𝒙𝟐 − 𝟏
𝒙 − 𝟐 (𝒙 + 𝟏)(𝒙 − 𝟏) (𝒙 − 𝟏)
× 𝟐
=
×
=
𝒙 + 𝟏 𝒙 − 𝟐𝒙 𝒙 + 𝟏
𝒙(𝒙 − 𝟐)
𝒙

Examples 65:

𝒑−𝒒 𝒑+𝒒 𝒑−𝒒 𝒑+𝒒 𝒑−𝒒
×
=
×
=
𝒑+𝒒
𝟐𝒃
𝒑+𝒒
𝟐𝒃
𝟐𝒃

Examples 66:
Factorization

𝒙𝟐 + 𝟓𝒙
𝟏
𝒙(𝒙 + 𝟓)
𝟏
𝟏
× 𝟐
=
×
=
(𝒙 + 𝟓)𝟐 𝒙 + 𝟓
𝒙
𝒙 + 𝟏𝟎𝒙 + 𝟐𝟓
𝒙
Identity I
Examples 67:

(𝒙 + 𝟐)(𝒙 + 𝟏)
𝒙 + 𝟑𝒙 + 𝟐
𝟏
×
=
=𝟏
𝒙+𝟐
𝒙 + 𝟏 (𝒙 + 𝟐) × (𝒙 + 𝟏)
Examples 68:

𝟔𝒂
𝟒(𝒂 + 𝟓)
𝟔𝒂
𝟒(𝒂 + 𝟓) 𝟒
×
=
×
=
𝟑(𝒂 + 𝟓)
𝟐𝒂𝟐
𝟑(𝒂 + 𝟓)
𝟐𝒂𝟐
𝒂
12 | P a g e

HELMA NOTE

Operations with Algebraic and Rational Expressions

Dividing a Rational expression with another rational expression
Examples 69:

𝟕𝒙𝒚 𝟏𝟒𝒚
÷ 𝟐 =
𝟒𝒙𝟐
𝒙

𝟕𝒙𝒚
𝒙𝟐
𝒙 𝟏 𝒙
×
=
× =
𝟒𝒙𝟐 𝟏𝟒𝒚 𝟒 𝟐 𝟖

Examples 70:

𝟒𝒙𝟐
𝟑𝒙
÷ 𝟑=
𝟖𝒙𝒚 𝒚

𝟒𝒙𝟐 𝒚𝟑 𝒙 𝒚 𝒙𝒚
×
= × =
𝟖𝒙𝒚 𝟑𝒚 𝟐 𝟑
𝟔

Examples 71:
2x

(𝟐𝟖𝒙𝒚) (𝒙 − 𝟐)
𝟐𝟖𝒙𝒚
𝟏𝟒𝒚
÷
=
×
= 𝟐𝒙
(𝒙 − 𝟐) 𝒙 − 𝟐
(𝟏𝟒𝒚)
𝒙−𝟐

Examples 72:

𝒙 + 𝟑𝒙 + 𝟐 𝒙 + 𝟏 (𝒙 + 𝟐)(𝒙 + 𝟏) (𝒙 + 𝟓)
÷
=
×
= (𝒙 + 𝟓)
(𝒙 + 𝟐)
(𝒙 + 𝟏)
𝒙+𝟐
𝒙+𝟓

Examples 73:

Identity III

(𝒎 − 𝟕)(𝒎 + 𝟕) 𝒎(𝒎 + 𝟏)
𝒎𝟐 − 𝟒𝟗
𝒎+𝟕
÷ 𝟐
=
×
= 𝒎(𝒎 − 𝟕)
𝒎+𝟏
𝒎 +𝒎
𝒎+𝟏
𝒎+𝟕
factorization
Examples 74:

𝒂𝟐 + 𝟖𝒂 + 𝟏𝟓
𝒂𝟐 − 𝟐𝟓
𝒂𝟐 + 𝟖𝒂 + 𝟏𝟓 𝒂𝟐 − 𝟓𝒂 + 𝟒
÷ 𝟐
= 𝟐
×
=
𝒂𝟐 + 𝟐𝒂 − 𝟑
𝒂 − 𝟓𝒂 + 𝟒
𝒂 + 𝟐𝒂 − 𝟑
𝒂𝟐 − 𝟐𝟓

=
13 | P a g e

(𝒂 + 𝟑)(𝒂 + 𝟓) (𝒂 − 𝟒)(𝒂 − 𝟏) 𝒂 − 𝟒
×
=
(𝒂 + 𝟑)(𝒂 − 𝟏) (𝒂 + 𝟓)(𝒂 − 𝟓) 𝒂 − 𝟓

HELMA NOTE

Operations with Algebraic and Rational Expressions

Sum and Difference of Rational Expressions
To add or subtract two rational expressions with the same denominator
Examples 75:
Add numerators

𝟏
𝒙 − 𝟏 (𝟏) + (𝒙 − 𝟏)
𝒙
+
=
=
𝒚+𝟑 𝒚+𝟑
𝒚+𝟑
𝒚+𝟑
Common
denominator
Examples 76:

𝒙 + 𝟓 𝟐𝒙 − 𝟏 𝒙 + 𝟓 − (𝟐𝒙 − 𝟏) 𝒙 + 𝟓 − 𝟐𝒙 + 𝟏 −𝒙 + 𝟔
−
=
=
=
𝒙+𝟓
𝒙+𝟓
𝒙+𝟓
𝒙+𝟓
𝒙+𝟓
Common
denominator
Examples 77:

𝟑𝒙 + 𝟕 𝟐𝒙 − 𝟑 𝟑𝒙 + 𝟕 + 𝟐𝒙 − 𝟑 𝟓𝒙 + 𝟒
+
=
=
𝒙+𝟐
𝒙+𝟐
𝒙+𝟐
𝒙+𝟐

Examples 77:

𝟑𝒙 + 𝟕 𝟐𝒙 − 𝟑
−
=
𝒙+𝟐
𝒙+𝟐

𝟑𝒙 + 𝟕 − (𝟐𝒙 − 𝟑) 𝟑𝒙 + 𝟕 − 𝟐𝒙 + 𝟑) 𝒙 + 𝟏𝟎
=
=
𝒙+𝟐
𝒙+𝟐
𝒙+𝟐
Examples 78:

𝟐
𝒎 − 𝟑 𝟐 − (𝒎 − 𝟑) 𝟐 − 𝒎 + 𝟑) −𝒎 + 𝟓
−
=
=
=
𝒎+𝟐 𝒎+𝟐
𝒎+𝟐
𝒎+𝟐
𝒎+𝟐

14 | P a g e

HELMA NOTE

Operations with Algebraic and Rational Expressions

To add or subtract two rational expressions with unlike denominator
Examples 79:

𝟏
𝟏
𝟏(𝟒𝒃)
𝟏(𝟑𝒂)
𝟒𝒃 + 𝟑𝒂
+
=
+
=
𝟑𝒂 𝟒𝒃 𝟑𝒂(𝟒𝒃) 𝟒𝒃(𝟑𝒂)
𝟏𝟐𝒂𝒃
The LCD of the
fraction is 12ab
Examples 78:

𝟏
𝟐
𝟏+𝟐
𝟑
+ =
=
𝒂 − 𝟓 𝒂 𝒂(𝒂 − 𝟓) 𝒂(𝒂 − 𝟓)
The LCM of the fraction is

𝒂(𝒂 − 𝟓)
Examples 79:

𝟐
𝒙 − 𝟏 𝟐(𝒙 + 𝟒) + (𝒙 − 𝟏)(𝒙 + 𝟐)
+
=
(𝒙 + 𝟐)(𝒙 + 𝟒)
𝒙+𝟐 𝒙+𝟒

Examples 80:

𝒂𝟐 − 𝟐𝟎 𝒂 − 𝟐
𝒂𝟐 − 𝟐𝟎
𝒂 − 𝟐(𝒂 − 𝟐)
+
=
+
𝒂𝟐 − 𝟒
𝒂 + 𝟐 (𝒂 + 𝟐)(𝒂 − 𝟐) 𝒂 + 𝟐(𝒂 − 𝟐)
𝒂𝟐 − 𝟐𝟎 + 𝒂 − 𝟐(𝒂 − 𝟐) 𝒂𝟐 − 𝟐𝟎 + 𝒂𝟐 − 𝟒𝒂 + 𝟒
=
=
𝒂 + 𝟐(𝒂 − 𝟐)
𝒂 + 𝟐(𝒂 − 𝟐)
𝟐𝒂𝟐 − 𝟒𝒂 − 𝟏𝟔 𝟐(𝒂𝟐 − 𝟐𝒂 − 𝟖) 𝟐(𝒂𝟐 − 𝟐𝒂 − 𝟖)
=
=
=
𝒂 + 𝟐(𝒂 − 𝟐)
𝒂 + 𝟐(𝒂 − 𝟐)
𝒂 + 𝟐(𝒂 − 𝟐)
=

𝟐((𝒂 − 𝟒)(𝒂 + 𝟐)) 𝟐(𝒂 − 𝟒)
=
(𝒂 − 𝟐)
𝒂 + 𝟐(𝒂 − 𝟐)

15 | P a g e

HELMA NOTE

Operations with Algebraic and Rational Expressions

Examples 81:

𝟐𝒙𝟐 − 𝟏𝟔 𝒙 + 𝟒
𝟐𝒙𝟐 − 𝟏𝟔
𝒙 + 𝟒(𝒙 − 𝟐)
+
=
+
=
𝒙𝟐 − 𝟒
𝒙 + 𝟐 (𝒙 + 𝟐)(𝒙 − 𝟐) 𝒙 + 𝟐(𝒙 − 𝟐)
𝟐𝒙𝟐 − 𝟏𝟔 + 𝒙𝟐 + 𝟐𝒙 − 𝟖 𝟑𝒙𝟐 + 𝟐𝒙 − 𝟐𝟒
=
(𝒙 + 𝟐)(𝒙 − 𝟐)
(𝒙 + 𝟐)(𝒙 − 𝟐)

Examples 82:

Identity IV

𝟐
𝒙−𝟏
𝟐(𝒙 + 𝟒)
𝒙 − 𝟏(𝒙 + 𝟐) 𝟐(𝒙 + 𝟒) + (𝒙 − 𝟏)(𝒙 + 𝟐)
+
=
+
=
(𝒙 + 𝟐)(𝒙 + 𝟒)
𝒙 + 𝟐 𝒙 + 𝟒 𝒙 + 𝟐(𝒙 + 𝟒) 𝒙 + 𝟒(𝒙 + 𝟐)

=

𝟐𝒙 + 𝟖 + 𝒙𝟐 + (𝟐 − 𝟏)𝒙 + (−𝟏 × 𝟐) 𝟐𝒙 + 𝟖 + 𝒙𝟐 + 𝒙 − 𝟐
=
(𝒙 + 𝟐)(𝒙 + 𝟒)
(𝒙 + 𝟐)(𝒙 + 𝟒)
(𝒙 − 𝟐)(𝒙 + 𝟑)
𝒙𝟐 + 𝒙 − 𝟔
=
=
(𝒙 + 𝟐)(𝒙 + 𝟒)
(𝒙 + 𝟐)(𝒙 + 𝟒)

Examples 83:

𝒙𝟐
𝒚𝟐
𝒙𝟐 (𝒚 − 𝒙)
𝒚𝟐 (𝒙 − 𝒚)
𝒙𝟐 𝒚 − 𝒙𝟑 + 𝒚𝟐 𝒙 − 𝒚𝟑
+
=
+
=
𝒙 − 𝒚 𝒚 − 𝒙 𝒙 − 𝒚(𝒚 − 𝒙) 𝒚 − 𝒙(𝒙 − 𝒚)
𝒙 − 𝒚(𝒚 − 𝒙)
Examples 84:

𝟔
𝟒
𝟔
𝟒(𝟓) 𝟔 + 𝟐𝟎 𝟐𝟔
+ =
+
=
=
𝟓𝒙 𝒙 𝟓𝒙 𝒙(𝟓)
𝟓𝒙
𝟓𝒙
Examples 85:

𝟕
𝒙
𝟕
𝒙
+
=
+
𝒙𝟐 − 𝒙 − 𝟐 𝒙𝟐 + 𝟒𝒙 + 𝟑 (𝒙 − 𝟐)(𝒙 + 𝟏) (𝒙 + 𝟑)(𝒙 + 𝟏)
=

𝟕(𝒙 + 𝟑)
𝒙(𝒙 − 𝟐)
+
(𝒙 − 𝟐)(𝒙 + 𝟏)(𝒙 + 𝟑) (𝒙 + 𝟑)(𝒙 + 𝟏)(𝒙 − 𝟐)

16 | P a g e

HELMA NOTE

Operations with Algebraic and Rational Expressions

𝟕𝒙 + 𝟐𝟏 + 𝒙𝟐 − 𝟐𝒙
𝒙𝟐 + 𝟓𝒙 + 𝟐𝟏
=
=
(𝒙 − 𝟐)(𝒙 + 𝟏)(𝒙 + 𝟑) (𝒙 − 𝟐)(𝒙 + 𝟏)(𝒙 + 𝟑)

 For learning these examples read this book:
“Operations with Algebraic and Rational Expressions”

Algebraic Identities
Identity I:

(𝒙 + 𝒚)𝟐 = 𝒙𝟐 + 𝟐𝒙𝒚 + 𝒚𝟐

Identity II:

(𝒙 − 𝒚)𝟐 = 𝒙𝟐 − 𝟐𝒙𝒚 + 𝒚𝟐

Identity III: (𝒙 + 𝒚)(𝒙 − 𝒚) = 𝒙𝟐 − 𝒚𝟐
𝟐

Identity IV: (𝒙 + 𝒂)(𝒙 + 𝒃) = 𝒙𝟐 + (𝒂 + 𝒃) 𝒙

17 | P a g e

+ (𝒂𝒃)



---