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Math 8 Lesson 3 - 2nd Quarter
- 2. Similar to adding and subtracting fractions with like or same denominators, combining rational expressions with like denominators is done by adding and/or subtracting the numerators and copying the common denominator. Let a, b, and c be nonzero polynomials. 𝒂 𝒄 + 𝒃 𝒄 = 𝒂 + 𝒃 𝒄 𝒂𝒏𝒅 𝒂 𝒄 − 𝒃 𝒄 = 𝒂 − 𝒃 𝒄 𝒘𝒉𝒆𝒓𝒆 𝒄 ≠ 𝟎.
- 3. 𝒙 + 𝟐 𝟑 + 𝒙 − 𝟒 𝟑 = 𝟑 𝒙 + 𝟐 + 𝒙 − 𝟒 = 𝟑 𝟐𝒙− 𝟐
- 4. 𝒚 𝟐 + 𝟏𝟎𝟎 𝐲 + 𝟏𝟎 + 𝟐𝟎𝒚 𝐲 + 𝟏𝟎 = 𝒚 + 𝟏𝟎 𝒚 𝟐 + 𝟏𝟎𝟎+ 𝟐𝟎𝒚 = 𝒚 + 𝟏𝟎 (𝒚 + 𝟏𝟎)(𝒚 + 𝟏𝟎) = 𝒚 + 𝟏𝟎
- 5. 𝒘 − 𝟏 𝒘 − 𝒘 − 𝟏 𝒘 = 𝒘 𝒘− 𝟏− 𝒘 +𝟏 = 𝒘 𝟎 = 𝟎
- 6. 𝒚 𝟐 + 𝟑𝟖 𝐲 − 𝟓 − 𝟏𝟑𝒚 − 𝟐 𝐲 − 𝟓 = 𝒚 − 𝟓 𝒚 𝟐 + 𝟑𝟖−𝟏𝟑𝒚 + 𝟐 = 𝒚 − 𝟓 (𝒚 − 𝟖) (𝒚 − 𝟓) = 𝒚 − 𝟖 = 𝒚 − 𝟓 𝒚 𝟐 −𝟏𝟑𝒚 +𝟒𝟎
- 7. These are cases where rational expressions are being combined do not have the same denominator. As with combining fractions with unlike denominators, the fractions are expressed in equal forms such that the denominators are the same. The denominator is the least common denominator or LCD. Let a, b, c and d be nonzero polynomials. Then, 𝒂 𝒄 + 𝒃 𝒅 = 𝒂𝒅 + 𝒃𝒄 𝒄𝒅 𝒂𝒏𝒅 𝒂 𝒄 − 𝒃 𝒅 = 𝒂𝒅 − 𝒃𝒄 𝒄𝒅
- 8. 𝒏 𝟒 − 𝟑𝒏 𝟓 = 𝟐𝟎 𝟓(𝒏) − 𝟒(𝟑𝒏) = − 𝟕𝒏 𝟐𝟎 = 𝟐𝟎 𝟓𝒏 − 𝟏𝟐𝒏 𝑳𝑪𝑫: 𝟒 = 𝟒. 𝟏 𝟓 = 𝟓. 𝟏 = 𝟏. 𝟒. 𝟓 = 𝟐𝟎
- 9. 𝒎 𝒎 + 𝟑 + 𝟐 𝒎 𝟐 + 𝟔𝒎 + 𝟗 = 𝒎 𝟐 + 𝟔𝒎 + 𝟗 (𝒎 + 𝟑)(𝒎)+(𝟏)(𝟐) = 𝒎 𝟐 + 𝟔𝒎 + 𝟗 𝒎 𝟐 + 𝟑𝒎 +𝟐 𝑳𝑪𝑫: 𝒎 𝟐 + 𝟔𝒎 + 𝟗 = (𝒎 + 𝟑)(𝒎 + 𝟑) 𝒎 + 𝟑 = 𝒎 + 𝟑 = (𝒎 + 𝟑)(𝒎 + 𝟑) = 𝒎 𝟐 + 𝟔𝒎 + 𝟗
- 10. 𝟑 𝒑 + 𝟐 − 𝟐 𝒑 − 𝟑 = 𝒑 𝟐 − 𝒑 − 𝟔 (𝒑 − 𝟑)(𝟑) −(𝒑 + 𝟐)(𝟐) = 𝒑 𝟐 − 𝒑 − 𝟔 𝟑𝒑 − 𝟗 −𝟐𝒑 − 𝟒 𝑳𝑪𝑫: 𝒑 − 𝟑 = 𝒑 − 𝟑 𝒑 + 𝟐 = 𝒑 + 𝟐 = (𝒑 + 𝟐)(𝒑 − 𝟑) = 𝒑 𝟐 − 𝒑 − 𝟔 = 𝒑 𝟐 − 𝒑 − 𝟔 𝒑 −𝟏𝟑
- 11. Seatwork 3.1
- 14. = 𝒂 + 𝟐𝒃 𝟑𝒂 − 𝒃 + 𝟒𝒂+ 𝟓𝒃 = 𝒂 + 𝟐𝒃 𝟕𝒂 + 𝟒𝒃 1. 2. = 𝒙 𝟐 − 𝟑 𝒙 + 𝟐 + 𝟔𝒙 − 𝟓 −𝟐𝒙 + 𝟕 = 𝒙 𝟐 − 𝟑 𝟓𝒙 + 𝟒
- 15. 3. 4. = 𝒂 + 𝟐𝒃 𝟑𝒂 − 𝒃 − 𝟒𝒂− 𝟓𝒃 = 𝒂 + 𝟐𝒃 −𝒂− 𝟔𝒃 = 𝒙 𝟐 − 𝟑 𝟔𝒙 − 𝟓 + 𝟐𝒙 − 𝟕 = 𝒙 𝟐 − 𝟑 𝟖𝒙 − 𝟏𝟐
- 16. 5. = 𝟑𝒎 𝟓 − 𝟕𝒏 𝟔𝒎 𝟐 − 𝟏 −𝟓𝒎 − 𝟒 +𝟑𝒎 − 𝟓 = 𝟑𝒎 𝟓 − 𝟕𝒏 − 𝟐𝒎 − 𝟏𝟎𝟔𝒎 𝟐
- 17. 6. = 𝟒𝟐𝒙 𝟑 𝒚 𝟑 𝟏𝟒𝒙𝒚 𝟐 (𝟔) + 𝟔𝒙 𝟐 𝒚(𝟐𝒙) + 𝟐𝟏 𝟑𝒙 𝟐 = 𝟒𝟐𝒙 𝟑 𝒚 𝟑 𝟖𝟒𝒙𝒚 𝟐 + 𝟏𝟐𝒙 𝟑 𝒚 𝑳𝑪𝑫: 𝟕𝒙𝒚 𝟐 = 𝟕. 𝒙. 𝒚. 𝒚 𝟑𝒙 𝟐 𝒚 = 𝟑. 𝒙. 𝒙. 𝒚 = 𝟑. 𝟕. 𝟐. 𝒙. 𝒙. 𝒙. 𝒚. 𝒚. 𝒚 = 𝟒𝟐𝒙 𝟑 𝒚 𝟑 = 𝟏𝟒𝒙 𝟐 𝒚 𝟑 𝟐𝟖𝒚 𝟐 + 𝟒𝒙 𝟐 𝒚 + 𝟐𝟏𝒙 𝟐𝒙 𝟑 𝒚 𝟑 = 𝟐. 𝒙. 𝒙. 𝒙. 𝒚. 𝒚. 𝒚 + 𝟔𝟑𝒙 𝟐 = 𝟑𝒙(𝟏𝟒𝒙 𝟐 𝒚 𝟑 ) 𝟑𝒙 𝟐𝟖𝒚 𝟐 + 𝟒𝒙 𝟐 𝒚 + 𝟐𝟏𝒙
- 18. 7. 𝑳𝑪𝑫: 𝒂 − 𝟐𝒃 = 𝒂 − 𝟐𝒃 𝒂 + 𝟐𝒃 = 𝒂 + 𝟐𝒃 = (𝒂 + 𝟐𝒃)(𝒂 − 𝟐𝒃) = 𝒂 𝟐 − 𝟒𝒃 𝟐 = 𝒂 𝟐 − 𝟒𝒃 𝟐 (𝒂 − 𝟐𝒃) (𝟑𝒂 − 𝒃) +(𝒂 + 𝟐𝒃)(𝟒𝒂 + 𝟓𝒃) = 𝒂 𝟐 − 𝟒𝒃 𝟐 𝟑𝒂 𝟐 − 𝟕𝒂𝒃 + 𝟐𝒃 𝟐 + 𝟒𝒂 𝟐 + 𝟏𝟑𝒂𝒃 + 𝟏𝟎𝒃 𝟐 = 𝒂 𝟐 − 𝟒𝒃 𝟐 𝟕𝒂 𝟐 + 𝟔𝒂𝒃 + 𝟏𝟐𝒃 𝟐
- 19. 8. 𝑳𝑪𝑫: 𝟐𝒙 𝟒 𝒚 𝟑 = 𝟐. 𝒙. 𝒙. 𝒙. 𝒙. 𝒚. 𝒚. 𝒚 𝟕𝒙𝒚 𝟐 = 𝟕. 𝒙. 𝒚. 𝒚 = 𝟕. 𝟐. 𝒙. 𝒙. 𝒙. 𝒙. 𝒚. 𝒚. 𝒚 = 𝟏𝟒𝒙 𝟒 𝒚 𝟑 = 𝟏𝟒𝒙 𝟒 𝒚 𝟑 (𝟐𝒙 𝟑 𝒚)(𝟐𝒙) − (𝟕)(𝟑𝒙 𝟐 ) = 𝟏𝟒𝒙 𝟒 𝒚 𝟑 𝟒𝒙 𝟒 𝒚 − 𝟐𝟏𝒙 𝟐 = 𝟏𝟒𝒙 𝟐 𝒚 𝟑 𝟒𝒙 𝟐 𝒚 − 𝟐𝟏 = 𝒙 𝟐 𝟏𝟒𝒙 𝟐 𝒚 𝟑 𝒙 𝟐 𝟒𝒙 𝟐 𝒚 − 𝟐𝟏
- 20. 9. 𝑳𝑪𝑫: 𝒂 + 𝟐𝒃 = 𝒂 + 𝟐𝒃 𝟐𝒂 − 𝟒𝒃 = 𝟐(𝒂 − 𝟐𝒃) = (𝟐)(𝒂 − 𝟐𝒃)(𝒂 + 𝟐𝒃) = 𝟐𝒂 𝟐 − 𝟖𝒃 𝟐 = 𝟐𝒂 𝟐 − 𝟖𝒃 𝟐 (𝒂 + 𝟐𝒃)(𝟒𝒂 + 𝟓𝒃)−(𝟐𝒂 − 𝟒𝒃)(𝟑𝒂 − 𝒃) = 𝟐𝒂 𝟐 − 𝟖𝒃 𝟐 𝟒𝒂 𝟐 + 𝟏𝟑𝒂𝒃 + 𝟏𝟎𝒃 𝟐 −(𝟔𝒂 𝟐 − 𝟏𝟒𝒂𝒃 + 𝟒𝒃 𝟐 ) = 𝟐𝒂 𝟐 − 𝟖𝒃 𝟐 −𝟐𝒂 𝟐 + 𝟐𝟕𝒂𝒃 + 𝟔𝒃 𝟐 = 𝟐𝒂 𝟐 − 𝟖𝒃 𝟐 𝟒𝒂 𝟐 + 𝟏𝟑𝒂𝒃 + 𝟏𝟎𝒃 𝟐 −𝟔𝒂 𝟐 + 𝟏𝟒𝒂𝒃 − 𝟒𝒃 𝟐
- 21. 10. 𝑳𝑪𝑫: 𝟏𝟖𝒎 𝟐 + 𝟑𝟎𝒎 + 𝟓𝟎 = (𝟐) (𝟗𝒎 𝟐 + 𝟏𝟓𝒎 + 𝟐𝟓) 𝟐𝟕𝒎 𝟑 − 𝟏𝟐𝟓 = (𝟑𝒎 − 𝟓)(𝟗𝒎 𝟐 + 𝟏𝟓𝒎 + 𝟐𝟓) = (𝟐)(𝟗𝒎 𝟐 + 𝟏𝟓𝒎 + 𝟐𝟓)(𝟑𝒎 − 𝟓)(𝟑𝒎 + 𝟓) = 𝟏𝟔𝟐𝒎 𝟒 + 𝟐𝟕𝟎𝒎 𝟑 − 𝟕𝟓𝟎𝒎 − 𝟏𝟐𝟓𝟎 𝟗𝒎 𝟐 − 𝟐𝟓 = (𝟑𝒎 − 𝟓)(𝟑𝒎 + 𝟓)
- 22. 10. 𝑳𝑪𝑫: = 𝟏𝟔𝟐𝒎 𝟒 + 𝟐𝟕𝟎𝒎 𝟑 − 𝟕𝟓𝟎𝒎 − 𝟏𝟐𝟓𝟎 = (𝟔𝒎 + 𝟏𝟎)(𝟔𝒎 − 𝟏)+(𝟗𝒎 𝟐 − 𝟐𝟓)(𝟒) −(𝟏𝟖𝒎 𝟐 + 𝟑𝟎𝒎 + 𝟓𝟎)(𝟐𝒎 𝟐 ) 𝟏𝟔𝟐𝒎 𝟒 + 𝟐𝟕𝟎𝒎 𝟑 − 𝟕𝟓𝟎𝒎 − 𝟏𝟐𝟓𝟎 = (𝟑𝟔𝒎 𝟐 + 𝟓𝟒𝒎 − 𝟏𝟎)+(𝟑𝟔𝒎 𝟐 − 𝟏𝟎𝟎) −(𝟑𝟔𝒎 𝟒 + 𝟔𝟎𝒎 𝟑 + 𝟏𝟎𝟎𝒎 𝟐) 𝟏𝟔𝟐𝒎 𝟒 + 𝟐𝟕𝟎𝒎 𝟑 − 𝟕𝟓𝟎𝒎 − 𝟏𝟐𝟓𝟎 = 𝟑𝟔𝒎 𝟐 + 𝟓𝟒𝒎 − 𝟏𝟎 + 𝟑𝟔𝒎 𝟐 − 𝟏𝟎𝟎 − 𝟑𝟔𝒎 𝟒 − 𝟔𝟎𝒎 𝟑 − 𝟏𝟎𝟎𝒎 𝟐 𝟏𝟔𝟐𝒎 𝟒 + 𝟐𝟕𝟎𝒎 𝟑 − 𝟕𝟓𝟎𝒎 − 𝟏𝟐𝟓𝟎 = −𝟑𝟔𝒎 𝟒 𝟏𝟔𝟐𝒎 𝟒 + 𝟐𝟕𝟎𝒎 𝟑 − 𝟕𝟓𝟎𝒎 − 𝟏𝟐𝟓𝟎 − 𝟔𝟎𝒎 𝟑 − 𝟐𝟖𝒎 𝟐 + 𝟓𝟒𝒎 − 𝟏𝟏𝟎 = (𝟐) (𝟐)(𝟖𝟏𝒎 𝟒 + 𝟏𝟑𝟓𝒎 𝟑 − 𝟑𝟕𝟓𝒎 − 𝟔𝟐𝟓) (−𝟏𝟖𝒎 𝟒 − 𝟑𝟎𝒎 𝟑 − 𝟏𝟒𝒎 𝟐 + 𝟐𝟕𝒎 − 𝟓𝟓) = 𝟖𝟏𝒎 𝟒 + 𝟏𝟑𝟓𝒎 𝟑 − 𝟑𝟕𝟓𝒎 − 𝟔𝟐𝟓 −𝟏𝟖𝒎 𝟒 − 𝟑𝟎𝒎 𝟑 − 𝟏𝟒𝒎 𝟐 + 𝟐𝟕𝒎 − 𝟓𝟓
- 23. Answer Skill Builder #1 only on page 54 and page 56, and answer Math Skill Milestone letters A and B pages 57 – 58. And Prepare for a QUIZ next meeting.