multiplication and division of RAE

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:
𝒂𝟓
𝟏𝟎
∙
𝟓
𝒂𝟑
=
𝒂𝟓
𝟐∙𝟓
∙
𝟓
𝒂∙𝒂𝟐
Divide or cancel out the
common factors.
Multiply the remaining
terms.
=
𝒂𝟐
𝟐
Simplify.
=
𝒂𝟑∙𝒂𝟐
𝟐∙𝟓
∙
𝟓
𝒂𝟑
=
𝟓𝒂𝟓
𝟏𝟎𝒂𝟑
=
𝒂𝟐
𝟐

6. Factor the numerator and
denominator completely
Example:
𝟑𝟎𝒃𝟐
𝟔𝒄
∙
𝟒𝒄𝟐
𝟏𝟓𝒃𝟒
=
𝟐∙𝟏𝟓∙𝒃𝟐
𝟐∙𝟑∙𝒄
∙
𝟐∙𝟐∙𝒄∙𝒄
𝟏𝟓∙𝒃𝟐∙𝒃𝟐
Divide or cancel out the
common factors.
Multiply the remaining
terms.
=
𝟒𝒄
𝟑𝒃𝟐
Simplify.
=
𝟐∙𝟏𝟓∙𝒃𝟐
𝟐∙𝟑∙𝒄
∙
𝟐∙𝟐∙𝒄∙𝒄
𝟏𝟓∙𝒃𝟐∙𝒃𝟐
=
𝟏𝟐𝟎𝒃𝟐𝒄𝟐
𝟗𝟎𝒃𝟒𝒄
=
𝟒
𝟑𝒃𝟐
𝒄

7. Factor the numerator and
denominator completely
Example:
𝟑𝒅
𝟑𝒇−𝟏𝟐
∙
𝟒𝒇−𝟏𝟔
𝟏𝟐𝒅𝟐
=
𝟑𝒅
𝟑(𝒇−𝟒)
∙
𝟒(𝒇−𝟒)
𝟑𝒅(𝟒𝒅)
Divide or cancel out the
common factors.
Multiply the remaining
terms.
=
𝟏
𝟑𝒅
Simplify.
=
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:
(𝒂𝟐−𝟗)
(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)
=
(𝒂𝟐−𝟗)
(𝒂−𝟑)
∙
𝟑𝒂
(𝒂+𝟑)

12. Factor the numerator and
denominator completely
Example: 𝒙𝟐
+ 𝟐𝒙 − 𝟖 ÷
𝒙𝟐−𝟐𝒙−𝟖
(𝒙𝟐−𝟏𝟔)
= (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)
= 𝒙𝟐
+ 𝟐𝒙 − 𝟖 ∙
(𝒙𝟐−𝟏𝟔)
𝒙𝟐−𝟐𝒙−𝟖