Polynomial arithmetic #7 - Questions and Answers

Question 1 The recursive formula of an arithmetic sequence is described below: 𝑎 1 = 19 ,   𝑎 𝑛 = 𝑎 𝑛 − 1 + 6 a 1 ​ =19,a n ​ =a n−1 ​ +6 Which represents the explicit form of this arithmetic sequence? A. 𝑎 𝑛 = 6 𝑛 + 13 a n ​ =6n+13 B. 𝑎 𝑛 = 6 𝑛 + 19 a n ​ =6n+19 C. 𝑎 𝑛 = 13 𝑛 + 6 a n ​ =13n+6 D. 𝑎 𝑛 = 19 𝑛 + 6 a n ​ =19n+6 Answer 1: Given: 𝑎 1 = 19 ,   𝑎 𝑛 = 𝑎 𝑛 − 1 + 6 a 1 ​ =19,a n ​ =a n−1 ​ +6 Here, the first term is 19, and each next term is obtained by adding 6 to the previous term. That is: 𝑎 1 = 19 a 1 ​ =19 𝑑 = 6 d=6 Now, the explicit equation is: 𝑎 𝑛 = 𝑎 1 + ( 𝑛 − 1 ) 𝑑 a n ​ =a 1 ​ +(n−1)d 𝑎 𝑛 = 19 + ( 𝑛 − 1 ) ⋅ 6 a n ​ =19+(n−1)⋅6 𝑎 𝑛 = 19 + 6 𝑛 − 6 a n ​ =19+6n−6 𝑎 𝑛 = 6 𝑛 + 13 a n ​ =6n+13 Thus, the correct explicit form is: A. 𝑎 𝑛 = 6 𝑛 + 13 a n ​ =6n+13 Question 2 How many times a particular number is a zero for a given polynomial? For example, in the polynomial 𝑓 ( 𝑥 ) = ( 𝑥 − 3 ) 4 ( 𝑥 − 5 ) 2 ( 𝑥 − 8 ) f(x)=(x−3) 4 (x−5) 2 (x−8), the zero 3 has multiplicity 4, 5 has multiplicity 2, and 8 has multiplicity 1. Although this polynomial has only three zeros, we say that it has seven zeros counting multiplicity. Answer 1: In your example, 3 3 is a zero of 𝑓 ( 𝑥 ) f(x) because 𝑓 ( 3 ) = 0 f(3)=0. That’s all it means to call something a zero of a polynomial. If 𝑓 ( 𝑥 ) f(x) is some number, and 𝑔 ( 𝑥 ) g(x) is a polynomial, we say 𝑛 n is a zero of 𝑔 ( 𝑛 ) g(n). Now, if we have two polynomials with different zeros, we know the polynomials are different. But how can we tell if: 𝑓 ( 𝑥 ) = ( 𝑥 − 2 ) ( 𝑥 − 3 ) f(x)=(x−2)(x−3) and 𝑔 ( 𝑥 ) = ( 𝑥 − 2 ) ( 𝑥 − 3 ) 2 g(x)=(x−2)(x−3) 2 are different? They both have 2 2 and 3 3 as their roots, so that won’t be quite enough to distinguish them. That’s where the multiplicity of a root comes in. Question 3 In how many ways can 8 people be seated in a row if: a) There are no restrictions on the seating arrangement. b) Persons 𝐴 A and 𝐵 B must sit next to each other. c) There are 4 men and 4 women, and no 2 men or 2 women can sit next to each other. d) There are 5 men and they must sit next to one another. e) There are 4 married couples, and each couple must sit together. Answer: Step 1 Given that there are 8 people sitting in a row, the number of people, 𝑛 = 8 n=8. Step 2: No Restrictions (a) The number of ways to arrange 8 people: 8 ! = 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 40 , 320 8!=8⋅7⋅6⋅5⋅4⋅3⋅2⋅1=40,320 Step 3: 𝐴 A and 𝐵 B Must Sit Together (b) Treat 𝐴 A and 𝐵 B as a single block. Now, there are 7 entities to arrange: 7 ! = 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 5 , 040 7!=7⋅6⋅5⋅4⋅3⋅2⋅1=5,040 Inside the block, 𝐴 A and 𝐵 B can swap places: 2 ! = 2 2!=2 Total arrangements: 7 ! ⋅ 2 ! = 5 , 040 ⋅ 2 = 10 , 080 7!⋅2!=5,040⋅2=10,080 Step 4: 4 Men, 4 Women, No Two of Same Gender Together (c) Arrange men and women alternately. Starting with men: 4 ! ⋅ 4 ! = 24 ⋅ 24 = 576 4!⋅4!=24⋅24=576 Multiply by 2 (starting with women is also valid): 2 ⋅ 576 = 1 , 152 2⋅576=1,152 Step 5: 5 Men Must Sit Together (d) Treat the 5 men as one block. Now, there are 4 entities to arrange: 4 ! = 24 4!=24 Within the block of 5 men, arrangements: 5 ! = 120 5!=120 Total arrangements: 4 ! ⋅ 5 ! = 24 ⋅ 120 = 2 , 880 4!⋅5!=24⋅120=2,880 Step 6: 4 Married Couples Sit Together (e) Treat each couple as a single block. There are 4 blocks to arrange: 4 ! = 24 4!=24 Within each block, there are 2 ! = 2 2!=2 arrangements. Total: 4 ! ⋅ 2 4 = 24 ⋅ 16 = 384 4!⋅2 4 =24⋅16=384