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Math 8 Lesson 2 - 2nd Quarter
- 2. The rules in multiplying and dividing rational expressions are the same as the rules in multiplying and dividing real valued fractions. Let a, b, c and d be polynomials. Then, 𝑎 𝑏 ∙ 𝑐 𝑑 = 𝑎𝑐 𝑏𝑑 , for all b and d not equal to zero; 𝑎 𝑏 ÷ 𝑐 𝑑 = 𝑎 𝑏 ∙ 𝑑 𝑐 = 𝑎𝑑 𝑏𝑐 , for all a, b, c and d not equal to zero.
- 4. The following steps are followed to multiply rational expressions. 1. Express the numerators and denominators as factors. 2. Cancel out common factors. 3. Perform multiplication of factors.
- 5. Example: Simplify the rational expressions. 1. Express the numerators and denominators as factors. 2. Cancel out common factors. 3. Perform multiplication of factors. 𝟑𝒙 𝟒 ∙ 𝟖 𝟑𝒙 𝟒 ∙ 𝟖 = 𝟑𝒙 𝟒 ∙ 𝟒. 𝟐 𝟑𝒙 𝟒 ∙ 𝟖 = 𝟑𝒙 𝟒 ∙ 𝟒. 𝟐 𝟑𝒙 𝟒 ∙ 𝟖 = 𝟑𝒙 𝟒 ∙ 𝟒. 𝟐 = 𝟑𝒙. 𝟐 = 𝟔𝒙 = 𝟔𝒙
- 6. Example: Simplify the rational expressions. 1. Express the numerators and denominators as factors. 2. Cancel out common factors. 3. Perform multiplication of factors. 𝒚 + 𝟏 𝒚 − 𝟒 ∙ 𝒚 − 𝟒 𝒚 − 𝟑 = 𝒚 + 𝟏 𝒚 − 𝟒 ∙ 𝒚 − 𝟒 𝒚 − 𝟑 = 𝒚 + 𝟏 𝒚 − 𝟒 ∙ 𝒚 − 𝟒 𝒚 − 𝟑 = 𝒚 + 𝟏 𝒚 − 𝟒 ∙ 𝒚 − 𝟒 𝒚 − 𝟑 = 𝒚 + 𝟏 𝟏 ∙ 𝟏 𝒚 − 𝟑 = 𝒚 + 𝟏 𝒚 − 𝟑 = 𝒚 + 𝟏 𝒚 − 𝟑
- 7. Example: Simplify the rational expressions. 1. Express the numerators and denominators as factors. 2. Cancel out common factors. 3. Perform multiplication of factors. 𝒘 𝟐 + 𝟐𝒘 + 𝟏 𝒘 𝟐 + 𝟏𝟎𝒘 + 𝟏𝟔 ∙ 𝒘 𝟐 − 𝒘 − 𝟔 𝒘 𝟐 − 𝟐𝒘 − 𝟑 𝒘 𝟐 + 𝟐𝒘 + 𝟏 𝒘 𝟐 + 𝟏𝟎𝒘 + 𝟏𝟔 ∙ 𝒘 𝟐 − 𝒘 − 𝟔 𝒘 𝟐 − 𝟐𝒘 − 𝟑 𝒘 𝟐 + 𝟐𝒘 + 𝟏 𝒘 𝟐 + 𝟏𝟎𝒘 + 𝟏𝟔 ∙ 𝒘 𝟐 − 𝒘 − 𝟔 𝒘 𝟐 − 𝟐𝒘 − 𝟑 = 𝒘 + 𝟏 𝒘 + 𝟏 𝒘 + 𝟖 𝒘 + 𝟐 ∙ 𝒘 − 𝟑 𝒘 + 𝟐 𝒘 − 𝟑 𝒘 + 𝟏 = 𝒘 + 𝟏 𝒘 + 𝟖 = 𝒘 + 𝟏 𝒘 + 𝟖 = 𝒘 + 𝟏 𝒘 + 𝟏 𝒘 + 𝟖 𝒘 + 𝟐 ∙ 𝒘 − 𝟑 𝒘 + 𝟐 𝒘 − 𝟑 𝒘 + 𝟏 = 𝒘 + 𝟏 𝒘 + 𝟏 𝒘 + 𝟖 𝒘 + 𝟐 ∙ 𝒘 − 𝟑 𝒘 + 𝟐 𝒘 − 𝟑 𝒘 + 𝟏
- 8. Example: Simplify the rational expressions. 1. Express the numerators and denominators as factors. 2. Cancel out common factors. 3. Perform multiplication of factors. 𝒛 𝟑 + 𝟖 𝒛 − 𝟐 ∙ 𝒛 𝟐 − 𝟒𝒛 + 𝟒 𝒛 𝟐 − 𝟒 𝒛 𝟑 + 𝟖 𝒛 − 𝟐 ∙ 𝒛 𝟐 − 𝟒𝒛 + 𝟒 𝒛 𝟐 − 𝟒 𝒛 𝟑 + 𝟖 𝒛 − 𝟐 ∙ 𝒛 𝟐 − 𝟒𝒛 + 𝟒 𝒛 𝟐 − 𝟒 = 𝒛 + 𝟐 𝒛 𝟐 − 𝟐𝒛 + 𝟒 𝒛 − 𝟐 ∙ 𝒛 − 𝟐 𝒛 − 𝟐 𝒛 − 𝟐 𝒛 + 𝟐 = 𝒛 𝟐 − 𝟐𝒛 + 𝟒 = 𝒛 𝟐 − 𝟐𝒛 + 𝟒 = 𝒛 + 𝟐 𝒛 𝟐 − 𝟐𝒛 + 𝟒 𝒛 − 𝟐 ∙ 𝒛 − 𝟐 𝒛 − 𝟐 𝒛 − 𝟐 𝒛 + 𝟐 = 𝒛 + 𝟐 𝒛 𝟐 − 𝟐𝒛 + 𝟒 𝒛 − 𝟐 ∙ 𝒛 − 𝟐 𝒛 − 𝟐 𝒛 − 𝟐 𝒛 + 𝟐
- 10. Simplify the rational expressions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 .
- 12. 1. 2. 3. 4. 5. = 𝟓𝟗𝐧 𝟗𝟗 ∙ 𝟖𝟎 𝟑𝟑𝐧 = 𝟓𝟗(𝟖𝟎) 𝟗𝟗(33) = 𝟒𝟕𝟐𝟎 𝟑𝟐𝟔𝟕 = 𝟑. 𝟑𝟏 𝟑. 𝟕𝐧 ∙ 𝟐. 𝟏𝟕𝐧 𝟑. 𝟏𝟕𝐧 = 𝟑𝟏(𝟐) 𝟕𝐧(3) = 𝟔𝟐 𝟐𝟏𝐧 = 𝟖𝟒 𝟑 ∙ 𝟑(𝟏𝟔𝐱) 𝟗𝟓 = 𝟖𝟒(𝟏𝟔𝐱) 𝟗𝟓 = 𝟏𝟑𝟒𝟒𝐱 𝟗𝟓 = 𝟔(𝐫 + 𝟐) 𝟒. 𝟓 ∙ 𝟒𝐫 𝟔(𝐫 + 𝟐)= 𝐫 𝟓 = 𝐫 𝟓 = 𝟐(𝐩 + 𝟔) 𝟒 ∙ 𝐩 − 𝟑 𝟐(𝐩 − 𝟑) = 𝐩 + 𝟔(𝟏) 𝟒(1) = 𝐩 + 𝟔 𝟒
- 13. 6. 7. = (𝐱 − 𝟓)(𝐱 − 𝟓) 𝟏𝟎(𝐱 − 𝟏𝟎) ∙ 𝐱 − 𝟏𝟎 −𝟗(𝐱 − 𝟓) = 𝐱 − 𝟓 𝟏𝟎(−𝟗) = 𝐱 − 𝟓 −𝟗𝟎 = 𝟖(𝐯 − 𝟕) 𝟖(𝐯 + 𝟔) ∙ (𝐯 + 𝟑)(𝐯 + 𝟔) 𝟖𝐯(𝐯 + 𝟑) = (𝐯 − 𝟕)(𝟏) (𝟖𝐯)(𝟏) = 𝐯 − 𝟕 𝟖𝐯
- 14. 8. 9. = 𝐦 + 𝟏 𝟑(𝐦 − 𝟓) ∙ 𝟖(𝐦 − 𝟏𝟎) (𝐦 − 𝟏𝟎)(𝐦 + 𝟏) = 𝟖(𝟏) 𝟑(𝐦 − 𝟓) = 𝟖 𝟑𝐦 − 𝟏𝟓 = 𝟗𝐫 𝟐(𝐫 − 𝟔) 𝟗𝐫(𝐫 + 𝟓) ∙ 𝟗𝐫(𝐫 + 𝟏) 𝟗𝐫 𝟐 (𝐫 − 𝟔) = (𝐫 + 𝟏)(𝟏) 𝐫 + 𝟓 = 𝐫 + 𝟏 𝐫 + 𝟓
- 15. 10 . = 𝟑𝐱 𝟐(𝟏𝟓) 𝐱 − 𝟗 ∙ (𝐱 − 𝟗)(𝐱 + 𝟒) 𝟑𝐱 𝟐 (𝐱 + 𝟒) = 𝟏𝟓(𝟏) 𝟏 = 𝟏𝟓
- 17. 5. Perform multiplication of factors. The following are the steps in dividing rational expressions: 1. Get the reciprocal of the divisor. 2. Change the operation to multiplication. 3. Express the numerators and denominators as factors. 4. Cancel out the common factors.
- 18. 5. Perform multiplication of factors. 1. Get the reciprocal of the divisor. 2. Change the operation to multiplication. 3. Express the numerators and denominators as factors. 4. Cancel out the common factors. Example: Simplify the rational expressions. − 𝟒𝒙 𝟑 ÷ 𝟏𝟔 𝟏 𝟏𝟔 − 𝟒𝒙 𝟑 ∙ 𝟏 𝟒. 𝟒 − 𝟒𝒙 𝟑 ∙ 𝟏 𝟏𝟔 − 𝟒𝒙 𝟑 ∙ 𝟏 𝟒. 𝟒 = − 𝒙. 𝟏 𝟑. 𝟒 = − 𝒙 𝟏𝟐 = − 𝒙 𝟏𝟐
- 19. 5. Perform multiplication of factors. 1. Get the reciprocal of the divisor. 2. Change the operation to multiplication. 3. Express the numerators and denominators as factors. 4. Cancel out the common factors. Example: Simplify the rational expressions. 𝒚 + 𝟏 𝒚 − 𝟒 ÷ 𝒚 + 𝟓 𝒚 − 𝟒 𝒚 − 𝟒 𝒚 + 𝟓 𝒚 + 𝟏 𝐲 − 𝟒 ∙ 𝒚 − 𝟒 𝐲 + 𝟓 𝒚 + 𝟏 𝐲 − 𝟒 ∙ 𝒚 − 𝟒 𝐲 + 𝟓 𝒚 + 𝟏 𝐲 − 𝟒 ∙ 𝒚 − 𝟒 𝐲 + 𝟓 = 𝒚 + 𝟏(𝟏) 𝟏(𝒚 + 𝟓) = 𝒚 + 𝟏 𝒚 + 𝟓 = 𝒚 + 𝟏 𝒚 + 𝟓
- 20. 5. Perform multiplication of factors. 1. Get the reciprocal of the divisor. 2. Change the operation to multiplication. 3. Express the numerators and denominators as factors. 4. Cancel out the common factors. Example: Simplify the rational expressions. 𝒘 𝟐 + 𝟓𝒘 + 𝟔 𝒘 𝟐 − 𝟒 ÷ 𝒘 𝟐 + 𝒘 − 𝟔 𝒘 − 𝟐 𝒘 − 𝟐 𝒘 𝟐 + 𝒘 − 𝟔 (𝒘 + 𝟑)(𝒘 + 𝟐) 𝒘 𝟐 + 𝟓𝒘 + 𝟔 𝒘 𝟐 − 𝟒 ∙ 𝒘 − 𝟐 𝒘 𝟐 + 𝒘 − 𝟔 (𝒘 + 𝟑)(𝒘 + 𝟐) 𝐰 − 𝟐 (𝐰 + 𝟐) ∙ 𝒘 − 𝟐 (𝐰 + 𝟑)(𝐰 − 𝟐) = 𝟏 (𝟏)(𝒘 − 𝟐) = 𝟏 𝒘 − 𝟐 = 𝟏 𝒘 − 𝟐 (𝒘 − 𝟐)(𝒘 + 𝟐) ∙ (𝒘 − 𝟐) (𝒘 + 𝟑)(𝒘 − 𝟐)
- 21. 5. Perform multiplication of factors. 1. Get the reciprocal of the divisor. 2. Change the operation to multiplication. 3. Express the numerators and denominators as factors. 4. Cancel out the common factors. Example: Simplify the rational expressions. 𝒙 𝟑 − 𝟏 𝒙 𝟐 − 𝟐𝒙 + 𝟏 ÷ 𝒙 𝟐 + 𝒙 + 𝟏 𝒙 − 𝟏 𝒙 − 𝟏 𝒙 𝟐 + 𝒙 + 𝟏 (𝒙 − 𝟏)(𝒙 𝟐 + 𝒙 + 𝟏) 𝒙 𝟑 − 𝟏 𝒙 𝟐 − 𝟐𝒙 + 𝟏 ∙ 𝒙 − 𝟏 𝒙 𝟐 + 𝒙 + 𝟏 (𝒙 − 𝟏)(𝒙 𝟐 + 𝒙 + 𝟏) (𝒙 − 𝟏)(𝒙 − 𝟏) ∙ 𝒙 − 𝟏 𝒙 𝟐 + 𝒙 + 𝟏 = 𝟏 𝟏 = 𝟏 = 𝟏 (𝒙 − 𝟏)(𝒙 − 𝟏) ∙ (𝒙 − 𝟏) (𝒙 𝟐 + 𝒙 + 𝟏)
- 23. Simplify the rational expressions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 .
- 25. 1. 2. 3. 4. 5. 𝟐 = 𝟕 ∙ 𝟖𝒙 𝟐 𝟏𝟖 𝟐 = 𝟕 ∙ 𝟖𝒙 𝟐 𝟗. 𝟐 𝟖𝒙 𝟐 = 𝟕. 𝟗 = 𝟖𝒙 𝟐 𝟔𝟑 𝟕 = 𝟏𝟖 ∙ 𝟗𝒂 𝟔 𝟕 = 𝟗. 𝟐 ∙ 𝟗𝒂 𝟔 𝟕. 𝒂 = 𝟐. 𝟔 = 𝟕𝒂 𝟏𝟐 𝟓 = 𝟐𝟎 ∙ 𝟑 𝟓𝒙 𝟓 = 𝟐𝟎 ∙ 𝟑 𝟓. 𝒙 𝟑 = 𝟐𝟎. 𝒙 = 𝟑 𝟐𝟎𝒙 𝟒𝒏 = 𝒏 − 𝟔 ∙ 𝟖𝒏 − 𝟒𝟖 𝟒𝒏 𝟒𝒏 = 𝒏 − 𝟔 ∙ 𝟖(𝒏 − 𝟔) 𝟒𝒏 𝟖. 𝟏 = 𝟏 = 𝟖 𝟕𝒂 𝟐 = 𝟕𝐚 𝟑 + 𝟓𝟔𝐚 𝟐 ∙ 𝒂 𝟐 + 𝟕𝒂 − 𝟖 𝟐 𝟕𝒂 𝟐 = 𝟕𝐚 𝟐 (𝐚 + 𝟖) ∙ (𝒂 + 𝟖)(𝒂 − 𝟏) 𝟐 (𝒂 − 𝟏)(𝟏) = 𝟐. 𝟏 = 𝒂 − 𝟏 𝟐
- 26. 6. 7. 𝒃 𝟐 − 𝟐𝒃 − 𝟏𝟓 = 𝟖𝒃 + 𝟐𝟎 ∙ 𝟒𝒃 + 𝟏𝟎 𝟐 (𝒃 − 𝟓)(𝒃 + 𝟑) = 𝟒(𝟐𝒃 + 𝟓) ∙ 𝟐(𝟐𝒃 + 𝟓) 𝟐 (𝒃 − 𝟓)(𝒃 + 𝟑) = 𝟒. 𝟏 = 𝒃 𝟐 − 𝟐𝒃 − 𝟏𝟓 𝟒 𝟏𝟎𝒃 𝟐 + 𝟒𝟐𝒃 + 𝟑𝟔 = 𝟔𝒃 𝟐 − 𝟐𝒃 − 𝟔𝟎 ∙ 𝟑𝒃 𝟐 − 𝟏𝟑𝒃 + 𝟏𝟎 𝟒𝟎𝒃 + 𝟒𝟖 (𝟓𝒃 + 𝟔)(𝟐𝒃 + 𝟔) = (𝟑𝒃 − 𝟏𝟎)(𝟐𝒃 + 𝟔) ∙ (𝟑𝒃 − 𝟏𝟎)(𝒃 − 𝟏) 8(𝟓𝒃 + 𝟔) (𝒃 − 𝟏)(𝟏) = 𝟖 = 𝒃 − 𝟏 𝟖
- 27. 8. 9. 𝟔𝒑 + 𝟐𝟕 = 𝟏𝟖𝒑 𝟐 + 𝟑𝟔𝒑 ∙ 𝟐𝒑 + 𝟒 𝟏𝟔𝒑 + 𝟕𝟐 (𝟑)(𝟐𝒑 + 𝟗) = 𝟔. 𝟑𝒑(𝒑 + 𝟐) ∙ 𝟐(𝒑 + 𝟐) 𝟖(𝟐𝒑 + 𝟗) 𝟏 = 𝟑𝒑. 𝟖 = 𝟏 𝟐𝟒𝒑 𝟏𝟎𝒙 𝟐 − 𝟐𝟖𝒙 + 𝟏𝟔 = 𝟐𝒙 − 𝟒 ∙ 𝟓𝒙 𝟐 − 𝟒𝟏𝒙 + 𝟖 𝟐𝟓𝒙 𝟐 − 𝟐𝟓𝒙 + 𝟒 (𝟓𝒙 − 𝟒)(𝟐𝒙 − 𝟒) = 𝟐𝒙 − 𝟒 ∙ (𝟓𝒙 − 𝟏)(𝒙 − 𝟖) (𝟓𝒙 − 𝟏)(𝟓𝒙 − 𝟒) (𝟏)(𝒙 − 𝟖) = 𝟏 = 𝒙 − 𝟖
- 28. 10 . 𝟑𝒙 𝟐 − 𝟐𝟓𝒙 − 𝟏𝟖 = 𝟐𝟕𝒙 + 𝟏𝟖 ∙ 𝟓𝒙 𝟐 − 𝟑𝟑𝒙 + 𝟏𝟖 𝟓𝒙 − 𝟑 (𝟑𝒙 + 𝟐)(𝒙 − 𝟗) = 𝟗(𝟑𝒙 + 𝟐) ∙ (𝟓𝒙 − 𝟑)(𝒙 − 𝟔) 𝟓𝒙 − 𝟑 (𝒙 − 𝟗)(𝒙 − 𝟔) = 𝟗. 𝟏 = 𝒙 𝟐 − 𝟏𝟓𝒙 + 𝟓𝟒 𝟗
- 30. 𝟏. 𝒙 + 𝟏 𝟑 𝒙 + 𝟏 𝟕 𝟐. 𝟐𝒏 − 𝟑 𝟑(𝟑 − 𝟐𝒏) 𝟑. 𝟖𝒙 𝟐 𝟏𝟐𝒙 𝟑 𝟒. 𝒂 𝟐 + 𝟔𝒂 𝒂𝒄 + 𝟔𝒄 𝟓. 𝟔𝒙 𝟑 (𝟐 − 𝒙) 𝟏𝟐𝒙 𝟒(𝟐 − 𝒙) 𝟔. 𝒙 𝟐 + 𝟔𝒙 + 𝟗 𝒙 𝟐 + 𝟐𝒙 − 𝟑 𝟕. 𝟖𝒙 𝟐 − 𝟏𝟒𝒙 + 𝟔 𝟒𝒙 − 𝟑 𝟖. 𝟑𝒙 𝟑 + 𝟏𝟖𝒙 𝟐 − 𝟐𝟏𝒙 𝒙 𝟑 + 𝟓𝒙 𝟐 − 𝟏𝟒 𝟗. 𝟗 − 𝒚 𝟐 𝒚 𝟐 + 𝟐𝒚 − 𝟏𝟓 𝟏𝟎. 𝒙 𝟐 − 𝟑𝒙 − 𝟏𝟎 𝒙 𝟐 − 𝟒𝒙 − 𝟓
- 31. 𝟏. 𝟖𝒙 𝟐 𝟗 ∙ 𝟗 𝟐 𝟐. 𝟗𝒏 𝟐𝒏 ∙ 𝟕 𝟓𝒏 𝟑. 𝟔𝒙 𝟐 𝟒 ∙ 𝟔 𝟓 𝟒. 𝟕 𝒎 − 𝟔 𝒎 − 𝟔 ∙ 𝟓𝒎 𝟕𝒎 − 𝟓 𝟒𝟗𝒎 − 𝟑𝟓 𝟓. 𝒙 𝟐 + 𝟏𝟏𝒙 + 𝟐𝟒 𝟔𝒙 𝟑 + 𝟏𝟖𝒙 ∙ 𝟔𝒙 𝟑 + 𝟔𝒙 𝟐 𝒙 𝟐 + 𝟓𝒙 − 𝟐𝟒 𝟏. 𝟖𝒙 𝟑𝒙 ÷ 𝟒 𝟕 𝟐. 𝟏𝟎𝒑 𝟓 ÷ 𝟖 𝟏𝟎 𝟑. 𝟕 𝟏𝟎 𝒏 + 𝟑 ÷ 𝒏 − 𝟐 𝒏 + 𝟑 𝐧 − 𝟐 𝟒. 𝟗 𝒃 𝟐 − 𝒃 − 𝟏𝟐 ÷ 𝒃 − 𝟓 𝒃 𝟐 − 𝒃 − 𝟏𝟐 𝟓. 𝟐𝒓 𝒓 + 𝟔 ÷ 𝟐𝒓 𝟕𝒓 + 𝟒𝟐