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1. Grade 8 – Mathematics
Quarter I
MULTIPLICATION AND DIVISION OF
RATIONAL ALGEBRAIC EXPRESSIONS

2. 1. recall multiplication and division
of fractions; and
2. multiply and add rational
algebraic expressions.

3. How do we multiply fractions?
𝟐
𝟑
∙
𝟑
𝟒
1. Multiply the numerators and denominators.
=
𝟔
𝟏𝟐
or
𝟏
𝟐
𝒂
𝒃
∙
𝒄
𝒅
=
𝒂𝒄
𝒃𝒅
where b ≠ 0 and d ≠ 0
𝟒
𝟓
∙
𝟐
𝟒
=
𝟖
𝟐𝟎
or
𝟐
𝟓
𝟐
𝟏
𝟏
𝟏
𝟏
𝟏

4. MULTIPLICATION OF RATIONAL EXPRESSIONS
𝑷
𝑸
∙
𝑹
𝑺
=
𝑷𝑹
𝑸𝑺
where P, Q, R and S are
polynomials; Q ≠ 0 and S ≠ 0
1. Factor the numerator and denominator
completely.
2. Divide or cancel out common factors.
3. Multiply the remaining terms.
4. Simplify, if possible.

5. Factor the numerator and
denominator completely
Example:
𝒂 𝟓
𝟏𝟎
∙
𝟓
𝒂 𝟑
=
𝒂 𝟓
𝟐∙𝟓
∙
𝟓
𝒂∙𝒂 𝟐
Divide or cancel out the
common factors.
Multiply the remaining
terms.
=
𝒂 𝟐
𝟐Simplify.
=
𝒂 𝟑∙𝒂 𝟐
𝟐∙𝟓
∙
𝟓
𝒂 𝟑
=
𝟓𝒂 𝟓
𝟏𝟎𝒂 𝟑
=
𝒂 𝟐
𝟐

6. Factor the numerator and
denominator completely
Example:
𝟑𝟎𝒃 𝟐
𝟔𝒄
∙
𝟒𝒄 𝟐
𝟏𝟓𝒃 𝟒
=
𝟐∙𝟏𝟓∙𝒃 𝟐
𝟐∙𝟑∙𝒄
∙
𝟐∙𝟐∙𝒄∙𝒄
𝟏𝟓∙𝒃 𝟐∙𝒃 𝟐
Divide or cancel out the
common factors.
Multiply the remaining
terms.
=
𝟒𝒄
𝟑𝒃 𝟐Simplify.
=
𝟐∙𝟏𝟓∙𝒃 𝟐
𝟐∙𝟑∙𝒄
∙
𝟐∙𝟐∙𝒄∙𝒄
𝟏𝟓∙𝒃 𝟐∙𝒃 𝟐
=
𝟏𝟐𝟎𝒃 𝟐 𝒄 𝟐
𝟗𝟎𝒃 𝟒 𝒄
=
𝟒
𝟑 𝒃 𝟐
𝒄

7. Factor the numerator and
denominator completely
Example:
𝟑𝒅
𝟑𝒇−𝟏𝟐
∙
𝟒𝒇−𝟏𝟔
𝟏𝟐𝒅 𝟐
=
𝟑𝒅
𝟑(𝒇−𝟒)
∙
𝟒(𝒇−𝟒)
𝟑𝒅(𝟒𝒅)
Divide or cancel out the
common factors.
Multiply the remaining
terms.
=
𝟏
𝟑𝒅Simplify.
=
𝟑𝒅
𝟑(𝒇−𝟒)
∙
𝟒(𝒇−𝟒)
𝟑𝒅(𝟒𝒅)
𝟒 ∙ 𝒅

8. Factor the numerator and
denominator completely
Example:
(𝒂+𝟐) 𝟑
(𝒂+𝟒)
(𝒂+𝟒) 𝟒
(𝒂+𝟐) 𝟑
=
(𝒂+𝟐)(𝒂+𝟐)(𝒂+𝟐)
(𝒂+𝟒)
(𝒂+𝟒)(𝒂+𝟒)(𝒂+𝟒)(𝒂+𝟒)
(𝒂+𝟐)(𝒂+𝟐)(𝒂+𝟐)
Divide or cancel out the
common factors.
Multiply the remaining
terms.
= (𝒂+𝟒)(𝒂+𝟒)(𝒂+𝟒)
𝟏
Simplify.
=
(𝒂+𝟐)(𝒂+𝟐)(𝒂+𝟐)
(𝒂+𝟒)
(𝒂+𝟒)(𝒂+𝟒)(𝒂+𝟒)(𝒂+𝟒)
(𝒂+𝟐)(𝒂+𝟐)(𝒂+𝟐)
=
(𝒂+𝟒) 𝟑
𝟏
= (𝒂 + 𝟒) 𝟑

9. How do we divide fractions?
𝒂
𝒃
÷
𝒄
𝒅
=
𝒂𝒅
𝒃𝒄
where b ≠ 0, c ≠ 0
and d ≠ 0
𝒂
𝒃
∙
𝒅
𝒄
=
𝟐
𝟑
÷
𝟒
𝟓
=
𝟏𝟎
𝟏𝟐
or=
𝟐
𝟑
∙
𝟓
𝟒
𝟓
𝟔

10. DIVISION OF RATIONAL EXPRESSIONS
𝑷
𝑸
÷
𝑹
𝑺
=
𝑷𝑺
𝑸𝑹
where P, Q, R and S
are polynomials; Q ≠ 0,
R ≠ 0 and S ≠ 0
1. Multiply the dividend and the reciprocal of the divisor.
2. Factor the numerator and denominator completely.
3. Divide or cancel out common factors.
4. Multiply the remaining terms.
5. Simplify, if possible.
𝑷
𝑸
∙
𝑺
𝑹
=

11. Factor the numerator and
denominator completely
Example:
(𝒂 𝟐−𝟗)
(𝒂−𝟑)
÷
(𝒂+𝟑)
𝟑𝒂
= (𝒂+𝟑)(𝒂−𝟑)
(𝒂−𝟑)
∙
𝟑𝒂
(𝒂+𝟑)
Divide or cancel out the
common factors.
Multiply the remaining
terms. = 𝟑𝒂
Simplify.
= (𝒂+𝟑)(𝒂−𝟑)
(𝒂−𝟑)
∙
𝟑𝒂
(𝒂+𝟑)
=
(𝒂 𝟐−𝟗)
(𝒂−𝟑)
∙
𝟑𝒂
(𝒂+𝟑)

12. Factor the numerator and
denominator completely
Example: 𝒙 𝟐
+ 𝟐𝒙 − 𝟖 ÷
𝒙 𝟐−𝟐𝒙−𝟖
(𝒙 𝟐−𝟏𝟔)
= (𝒙 + 𝟒)(𝒙 − 𝟐) ∙
(𝒙+𝟒)(𝒙−𝟒)
(𝒙−𝟒)(𝒙+𝟐)
Divide or cancel out the
common factors.
Multiply the remaining
terms. =
(𝒙−𝟐)(𝒙+𝟒) 𝟐
(𝒙+𝟐)
Simplify.
= (𝒙 + 𝟒)(𝒙 − 𝟐) ∙
(𝒙+𝟒)(𝒙−𝟒)
(𝒙−𝟒)(𝒙+𝟐)
= 𝒙 𝟐
+ 𝟐𝒙 − 𝟖 ∙
(𝒙 𝟐−𝟏𝟔)
𝒙 𝟐−𝟐𝒙−𝟖