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Q1:Math-Help-Services.com Introduction to Polynomials 3 Shannon MacLean Question 1 Question 3 Which of the following polynomials are written with their Indicate the degree of the polynomial given below. terms in the proper order? 1. 4x-1 2. x4 - 4x2 3. 2x+5x2 + 9×4 16x+4-x2 Question 4 Indicate the degree of the polynomial given below. Question 2 Which of the following polynomials are written with their terms in the proper order? 7-5x2 -12x -- 1. 3x2 -> + 5x4 + 10 2. 4x3 - x2 +9x-7 3. 5x3 -x+12 Question 5 indicate the degree of the polynomial given below. 9+5x4 http://math-help-services.org 20/04/2018 09:42:46 Page 1 of 3 Math-Help-Services.com 8 Question 6 Which of the following polynomials are written with their terms in the proper order? Question 8 Which of the following polynomials are written with their terms in the proper order? 1. x - x4 -x3 2. 7x-4x2 -2x4 3. 5x+ +9x2 - 8 1. x2 - 4 2. 5x2 - x-15x4 3. 81-x4 Question 7 Question 9 Which of the following polynomials are written with their terms in the proper order? Indicate the degree of the polynomial given below. 6x-5x3 http://math-help-services.org 20/04/2018 09:42:46 Page 2 of 3 1. 6x5 -3x4 -x3 2. 2x4 +7x- 422 - x3 3. 3x2 + 7x4 -12x6 9 Math-Help-Services.com Question 10 Which of the following polynomials are written with their terms in the proper order? 1. x3 -7x2 + 12x 2. x2 - 19+ 11x3 3. x+1 http://math-help-services.org 20/04/2018 09:42:46 Page 3 of 3 10 Math-Help-Services.com Introduction to Polynomials 3 Shannon MacLean - MTH-S42 Module 1 (16719) Answer Sheet Question 1 1 and 2 only Question 2 2 and 3 only Question 3 2 Question 4 2 Question 5 4 Question 6 3 only Question 7 1 only Question 8 1 only Question 9 3 Question 10 1 and 3 only - http://math-help-services.org 20/04/2018 09:42:46 Page 1 of 1See Answer
Q2:Math-Help-Services.com Introduction to Polynomials 1 Shannon MacLean Question 1 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 4 Identify the variable and the constant in the polynomial given below. 2-x2 +3x2 - 4x+2x +6+ 4x2 = ? 9x3 +5x2 - 5 Question 2 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 5 Combine the like terms of the polynomial given below and. identify the leading coefficient of the resulting polynomial. 2x2 -2 -3+2x+6+4x2 -10-2 = ? Question 3 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 6 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. 3x2 +5x+6-2x2 +2x-1 =? 2×3 -3x-2x3 - 1+3x+52 - 1 =? 9x3 -x-2x3 - 1+3x+5x2 - 5 =? http:/math-help-services.org 20/04/2018 09:39:56 Page 1 of 2 Math-Help-Services.com Question 7 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 9 Identify the variable and the constant in the polynomial given below. 4-3x2 +x+5-4x+ 7x2 = ? ₹ -4x+5 Question 8 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 10 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. ≥ +3x+2x3 +4+23 - 2x2 -3x+6 =? 2x3 +5% +2x-62 -1 =? http://math-help-services.org 20/04/2018 09:39:57 Page 2 of 2 Math-Help-Services.com Introduction to Polynomials 1 Shannon MacLean - MTH-S42 Module 1 (16719) Answer Sheet Question 1 6x2 - 2x+8 leading coefficient is 6 Question 2 7x3 + 5x2 + 2x-6 leading coefficient is 7 Question 3 5x2 -2 leading coefficient is 5 Question 4 variable is x constant is - 5 Question 5 -2x3 +6x2 +2x-7 leading coefficient is - 2 Question 6 x +7x+5 leading coefficient is 1 Question 7 4x2 - 3x+ 9 leading coefficient is 4 Question 8 4x3 - 2 + 10 leading coefficient is 4 Question 9 variable is x constant is 5 Question 10 2x3 - x2 +2x-1 leading coefficient is 2 http:/math-help-services.org 20/04/2018 09:39:57 Page 1 of 1 3See Answer
Q3:Question 2 Consider the gambler's ruin with draws: Alice starts with fa and Bob with £(m - a), and at each time step Alice wins £1 off Bob with probability p, loses £1 to Bob with probability q, and no money is exchanged with probability s, where p +q + 8 =1. We consider the case where Bob and Alice are equally matched, so p= q and s = 1- 2p. (We assume 0<p <1/2.) Let ry be Alice's ruin probability from the point she has Li. (a) By conditioning on the first step, explain why pri+1-(1-s)r.+pr ;- 1 = 0, and give appropriate boundary conditions. [2] (b) Solve this linear difference equation to find an expression for ry- [2] Let d, be the expected duration of the game from the point Alice has &i. (c) Explain why pd141 - (1-8)d, + pd1-1 =- 1, and give appropriate boundary conditions. [2] (d) Solve this linear difference equation to find an expression for d .. [2] (e) Compare your answer to parts (b) and (d) with those for the standard gambler's ruin problem with p = 1/2, and give reasons for the similarities or differences. [2]See Answer
Q4:College of Arts & Sciences Computer Science Department CS 3653 - Discrete Mathematics for Computer Science BA # 6 Chapter # 5 Max. Points # 10 QUESTIONS Pts Use the Principle of Mathematical Induction to prove: SN 1 Any integer amount of postage from 18 cents on up can be made from an infinite supply of 4-cent and 7-cent stamps. 2 Use the Principle of Mathematical Induction to prove: 1 + 5 + 9 + ... + (4n - 3) = n(2n - 1) for every positive integer n. 5 5 Spring - 2024 Page 1 of 1See Answer
Q5:Question 1 12 pts Let A = {1, 2, 3, 4}. Let the functions F, G, and H be given with domain and codomain A defined as F(1)=3,F(2)=2,F(3)=2,and F(4)=4; G(1) =1,G(2) =3,G(3) = 4, and G(4) = 2; H(1) =2, H(2) =4, H(3) = 1, and H(4) = 3. Find the following: (a) Fo G (b) Ho F (c) G o H (d) Fo Go H Edit View Insert Format Tools Table 12pt v Paragraph V B I U AV & VTV V 3º V V V : Time Attem 5 Da' Secol p 0 words </> Question 2 10 pts Given: f ={(a, 1), (b, 2), (c, 3), (d, 4)} g-1={(2,1),(3,2),(4,3), (1,4)} Verify the following: (g o f)-1 (x) = f-1 og-1(x) Edit View Insert Format Tools Table 12pt v V Paragraph BIUAV T2 > V V D V V : p 0 words > Question 3 10 pts If f : R\{-1} -> R\{1} is defined by f(x) = 1+x 1-x and g : R -> R is defined by g(x) = 1+22, then 2x prove that f o g(x) =(f(x))2. Edit View Insert Format Tools Table 12pt v Paragraph B I U A V & V TV 00 V > > ... p 0 words </> Question 4 8 pts Let f(x) = ax + b and g(x) = cx + d, where a, b, c, and d are constants. Determine necessary and sufficient conditions on the constants a, b, c, and d so that f og = go f. Edit View Insert Format Tools Table 12pt v V Paragraph B I U A V & V TV V V A VE V : p 0 words Question 5 9 pts Prove the following using the Pigeonhole Principle. (a) Prove that among any group of 27 English words, there are at least two words that start with the same letter. (b) Prove that among any group of 13 people, there are at least two people who have the same number of friends within the group, when each person does not know atleast one person. (c) In a group of 101 people, each person in the group knows a unique set of 10 languages. Prove that there are two people in the group who can communicate with each other in at least two different languages. Edit View Insert Format Tools Table 12pt Paragraph v B I U A V & V TV 20 V V D V : p 0 words 7 Question 6 16 pts For the following sequences find their closed AND recursive expressions. Be sure to add in all parts for both (a) (3, 7, 11, 15, ... ) (b) (2, 6, 12, 20, ... ) (c) (1, -2, 3, -4, ... ) (d) (1, 3, 6, 10, ... ) Edit View Insert Format Tools Table 12pt Paragraph V B I U A V & V TV 00 V V V V ... p 0 words </> 1 Question 7 12 pts Use and/or ][ notation to rewrite the following (a) (x+y+1)-(x+y+2)+(x+y+3) -... +(x+y+57) (b) (x+1)(x2 +2)(x3+3) ... (x20 +20) (c) 1(0+1) +2(0+1+2)+3(0+1+2+3) Edit View Insert Format Tools Table 12pt v V Paragraph BIUA > > T2 V 30 V V A > ilılı V : p 0 wordsSee Answer
Q6:1. Let A be the area between the x-axis, the curve y=x+, and x = 2 and x 5. Find a such that the line xa divides this area into two regions of equal area. 2. Find the volume of the solid of revolution when the area between y x and y =x³ between x=0 and x = 1 is rotated around the x-axis.See Answer
Q7:9) [5] The partial ordering relation S is shown with the Hasse diagram to the right. List the elements S. 20 10See Answer
Q8:8) [6] Consider the undirected graph to the right. a) List the vertices. b) List the edges. c) List the degree of each vertex in a table. d) Is it a simple graph? Why or why not?See Answer
Q9:7) [10] Let A = {1,2,3,4}. Define the relation R on A by aRb if and only if a > b - 1. a) List the elements of R. b) Use technology to create a digraph. Paste to the right. (6 points) You will earn half credit for a hand drawn picture. A nice free program: graphonline.ru/en/ c) Is R reflexive? Why or why not? d) Is R transitive? Why or why not?See Answer
Q10:5) [3] Let R and S be relations on Z where R = {(n,n) VnE Z} and S = {(1, 2) Vn E Z}. Hint: write out some of the terms of R and S for insight into the problem. a) Find RS. b) Find SR.See Answer
Q11:4) [5] Let R and S be relations on A = {1,2,3,4,5,6} such that R = {(1,3), (1,4), (2,2), (4,1),(5,2), (6,3)} and S = {(2,2), (2,4), (3,1),(4,1), (3,5)}. a) Find RS. b) Find SR.See Answer
Q12:3) [4] Let R be a relation from A = {2,4,6,8,10} to B = {-1,0,1} such that R = {(a, b) = A x BV a = 2b +6}and S be a relation from B = {-1,0,1} to C = {2,3,4} such that S = {(b, c) € B x C Vc - bisodd}. a) List the element of R. b) List the element of S.See Answer
Q13:2) [9] Let A = {2,4,6,8,10}. Let S be a relation on A defined by (a, b) ES if and only if a V b (a divides b). a) List the elements of S. b) Is S reflexive? Why or why not? c) Is S symmetric? Why or why not? d) Is Santisymmetric? Why or why not? e) Is Stransitive? Why or why not? f) Is Sa partial ordering? Why or why not? g) Is San equivalence relation? Why or why not?See Answer
Q14:1) [4] Let A = {2,4,6,8,10}. Let R be a relation on A defined by (a, b) E R if and only if a > b+ 1. Are the following elements of R? Why or why not? a) (4,4) b) (8,2) c) (2,10) d) (6,0)See Answer
Q15:4. (Kernel Ridge Regression) We assume a model of form y = f(x), where our task is to learn f, the regression function. Here, y is the response variable which we assume is an element of R, and the x are the Rd covariates or predictors. Pairs (x¹, y¹) € Rd x R are observed, and we seek f such that y = f(x¹) for i = 1,...,n. Given an RKHS H with kernel K, we can estimate f by solving an optimization problem over the RKHS, 1 f = arg min = [(y¹ − f(x¹))² + |||f|||· SEH 2 By the Representer Theorem, we know that f(-) = Σα;Κ(·,π΄). j=1 Let y = [y¹,,y"]¹₁a = [α₁,,an]¹ € R" and K € R"x" with Kij Then we obtain KB. a := arg min || - K3||2 + BER" Let K(x, 2) = (1 + Σ;_1 ª;²;)² be the polynomial kernel of degree 2. (a) What is the corresponding RKHS? (b) Write down the expression of f in this case. = K(x², x³).See Answer
Q16:4. Define a relation on the plane by setting (xo, yo) < (x1, y₁) if either yo − x² < y₁ − x², or yo – x² = y₁ − x² and x < x₁. Show that this is an order relation on the plane, and describe it geometrically.See Answer
Q17:2. Let C be a relation on a set A. If Ao C A, define the restriction of C to Ao to be the relation Cn (Ao x Ao). Show that the restriction of an equivalence relation is an equivalence relation.See Answer
Q18:FEB. 16 1. Let f A B. Let Ao C A and Bo C B. (a) Show that Ao C f¹(f(Ao)) and that equality holds if f is injective. (b) Show that f(f-¹ (Bo)) C Bo and that equality holds if f is surjective.See Answer
Q19:Problem 5. A perfect square is an integer which is also square of an integer. More formally, n E Z is a perfect square if n = a² for some a € Z. Prove or disprove the following two claims. a) If m and n are perfect squares, then the product mn is also a perfect square. b) If m and n are perfect squares, then the sum m + n is also a perfect square.See Answer
Q20:Problem 4. Let the universe for the variables in the following statement consist of all real numbers. Negate and simplify Vx+y[(lx] = [y]) → (y = ±x)].See Answer

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