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# Multiplication and Division of Rational Algebraic Expressions

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References:
Oronce, O. A., Mendoza, M.O. (2018), Grade 8 Mathematics: Exploring Math. Rex Publishing, Manila, Philippines.

Nivera, G. C. (2013), Grade 8 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.

For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE! 
https://tinyurl.com/y9muob6q

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References:
Oronce, O. A., Mendoza, M.O. (2018), Grade 8 Mathematics: Exploring Math. Rex Publishing, Manila, Philippines.

Nivera, G. C. (2013), Grade 8 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.

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### Multiplication and Division of Rational Algebraic Expressions

1. 1. Grade 8 – Mathematics Quarter I MULTIPLICATION AND DIVISION OF RATIONAL ALGEBRAIC EXPRESSIONS
2. 2. 1. recall multiplication and division of fractions; and 2. multiply and add rational algebraic expressions.
3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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