# Ghorpade's A Quick Review Of Commutative Algebra [Short Lecture] PDF

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6x Ϫ 9y Ϫ 8x ϩ 4y 17. Ϫ5n2 Ϫ 6n ϩ 7n2 ϩ 5n Ϫ 1 for n ϭ Ϫ6 18. 7 3. Ϫ4 Ϫ (Ϫ3) ϩ (Ϫ5) Ϫ 7 ϩ 10 19. Ϫ2xy Ϫ x ϩ 4y 4. 7 Ϫ 8 Ϫ 3 ϩ 4 Ϫ 9 Ϫ 4 ϩ 2 Ϫ 12 1 1 2 5. 5aϪ b Ϫ 3aϪ b ϩ 7aϪ b ϩ 1 3 2 3 6. (Ϫ6) и 3 Ϭ (Ϫ2) Ϫ 8 Ϭ (Ϫ4) 2 1 7. Ϫ (3 Ϫ 7) Ϫ (2 Ϫ 17) 2 5 8. [48 ϩ (Ϫ93)] ϩ (Ϫ49) 20. 4(n2 ϩ 1) Ϫ (2n2 ϩ 3) Ϫ 2(n2 ϩ 3) for n ϭ Ϫ4 9. 3(Ϫ2)3 ϩ 4(Ϫ2)2 Ϫ 9(Ϫ2) Ϫ 14 10. [2(Ϫ6) ϩ 5(Ϫ4)][Ϫ3(Ϫ4) Ϫ 7(6)] 11. [Ϫ2(Ϫ3) Ϫ 4(2)]5 12. Simplify 6x 2 Ϫ 3x Ϫ 7x 2 Ϫ 5x Ϫ 2 by combining similar terms. 13. Simplify 3(3n Ϫ 1) Ϫ 4(2n ϩ 3) ϩ 5(Ϫ4n Ϫ 1) by removing parentheses and combining similar terms.

3 3 3 Solution ΂ ΃ ΂ ΃ ΂ ΃ ΂ ΃ 1 2 1 1 8 5 Ϫ2 ϩ 4 Ϫ Ϫ (Ϫ5) Ϫ ϭ Ϫ2 ϩ Ϫ ϩ Ϫ 3 3 3 3 3 3 ΂ ΃ ΂ ΃ 7 8 5 ϭϪ ϩ Ϫ ϩ Ϫ 3 3 3 20 ϭϪ 3 Classroom Example Simplify 21 Ϭ (Ϫ3) ϩ 7( Ϫ2). EXAMPLE 10 Change to an improper fraction Simplify Ϫ24 Ϭ 4 ϩ 8(Ϫ5) Ϫ (Ϫ5)(3). 7(1 Ϫ (Ϫ4) ) ] . 6(6 Ϫ 7) ] . 5 Classroom Example Simplify: [5(Ϫ2) Ϫ 6(4) ] [ 4(Ϫ2) ϩ 7(1) ] EXAMPLE 12 Simplify [3(Ϫ7) Ϫ 2(9) ] [5(Ϫ7) ϩ 3(9) ] . , the airline sold 52 economy seats, 25 businessclass seats, 12 first-class seats, and there were 20 empty seats.

11. 7(4) ϩ 9(4) ϭ (7 ϩ 9)4 35. 7(2)3 ϩ 4(Ϫ2)3 12. (x ϩ 3) ϩ (Ϫ3) ϭ x ϩ [3 ϩ (Ϫ3)] 36. Ϫ4(Ϫ1)2 Ϫ 3(2)3 13. [(Ϫ14)(8)](25) ϭ (Ϫ14)[8(25)] 3 4 14. a ba b ϭ 1 4 3 For Problems 15–26, simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. (Objective 2) 15. 16. 17. 18. 36 ϩ (Ϫ14) ϩ (Ϫ12) ϩ 21 ϩ (Ϫ9) Ϫ 4 Ϫ37 ϩ 42 ϩ 18 ϩ 37 ϩ (Ϫ42) Ϫ 6 [83 ϩ (Ϫ99)] ϩ 18 [63 ϩ (Ϫ87)] ϩ (Ϫ64) 19. (25)(Ϫ13)(4) 3(Ϫ1)3 Ϫ 4(3)2 32. (Ϫ3)3 ϩ 32 34.