Questions and Answers #1 Polynomial Arithmetic

Questions and Answers Sheet 1 Polynomial Arithmetic Question #1 In how many ways can 8 people be seated in a row if a.) there are no restrictions on the seating arrangement (answer is 8! = 40,320), b.) persons A and B must sit next to each other? (Answer: 10,080), c.) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (Answer: 1,152), d.) there are 5 men and they must sit next to one another? (Answer: 2,880), e.) there are 4 married couples and each couple must sit together? (Answer: 384). I need to see the steps to solve questions 10b to 10e. I quickly looked through the corresponding chapter on combinatorial analysis and I went over topics such as the multinomial theorem, the basic principle of counting, the binomial theorem, and representing the number of possible combinations of n objects taken r at a time as ''n choose r''. Answer: Step 1 Given that there are 8 people sitting in a row. The number of people, n=8. Step 2 (a) To find the number of possible ways in which 8 people can be seated in a row, when there are no restrictions. The number of ways in which n people can be seated in a row, when there are no restrictions =n! Therefore, The number of ways =n! =8! 8×7×6×5×4×3×2×1 =40, 320 Step 3 (b) To find the number of possible ways in which 8 people can be seated in a row, when A and B must sit next to each other. Consider A and B be a single block and then proceed. Now, there are 6 people and one block consisting of A and B. Observe that there are 7 positions to permute with. The number of ways in which n people can be seated in a row, when there are no restrictions =n! Therefore, The number of ways =n! =n! 7×6×5×4×3×2×1 =5,040 Step 4 Inside the block there are A and B. The number of ways in which n people can be seated in a row, when there are no restrictions =n! Therefore, the number of ways in which A and B can be seated inside the block = 2! = 2 Then, the total number of ways in which 8 people can be seated in a row with A and B must sit together is The total number of ways = 2 × 5040 = 10,080 ways Question #2 Factorial, but with addition [duplicate] Is there a notation for addition form of factorial? 5! = 5 × 4 × 3 × 2 × 1 That's pretty obvious. But I'm wondering what I'd need to use to describe 5+4+3+2+1 like the factorial 5! way. Answer: It is called the n-th triangle number and it can be written as ( n +1 ) , as a binomial coefficient. 2 Question #3 To find: The real zeros of polynomial function form an arithmetic sequence 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 − 4𝑥 2 + 16𝑥 Answer: A ration expression is a fraction that is quotient of two polynomials. A rational function is defined by the two polynomial functions. A function f of the form 𝑓(𝑥) = 𝑝(𝑥) 𝑞(𝑥) is a rational function. Where, p(x) and q(x) are polynomial functions, with 𝑞(𝑥) ≠ 0. Calculation: The given polynomial function form an arithmetic sequence is 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 − 4𝑥 2 + 16𝑥 Here, the constant is 0. The above equation can be rewritten as 𝑓(𝑥) = 𝑥(𝑥 3 − 4𝑥 2 − 4𝑥 + 16) 𝑝 The possibilities for are ±1, ± 2, ± 4 and ±8 𝑞 Factoring the term (𝑥 3 − 4𝑥 2 − 4𝑥 + 16),we get (𝑥 3 − 4𝑥 2 − 4𝑥 + 16) = (𝑥 + 2)(𝑥 2 − 6𝑥 + 8) Factoring the term (𝑥 2 − 6𝑥 + 8), we get (𝑥 2 − 6𝑥 + 8) = (𝑥 − 2)(𝑥 − 4) Combining all the terms, we get (𝑥 3 − 4𝑥 2 − 4𝑥 + 16) = (𝑥 + 2)(𝑥 − 2)(𝑥 − 4) So, 𝑓(𝑥) = 𝑥(𝑥 3 − 4𝑥 2 − 4𝑥 + 16) = 𝑥(𝑥 + 2)(𝑥 − 2)(𝑥 − 4) Thus, the real zeros are -2, 0, 2, and 4. Question #4 Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q are the same. 𝑃(𝑥) = 3𝑥 4 − 5𝑥 3 + 𝑥 2 − 3𝑥 + 5 𝑄(𝑥) = (((3𝑥 − 5)𝑥 + 1)𝑥 − 3) 𝑥 + 5 Try to evalue P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial 𝑅(𝑥) = 𝑥 5 − 2𝑥 4 + 3𝑥 3 − 2𝑥 2 + 3𝑥 + 4 in ''nested'' form, like the polynomial Q. Use the nested form to find R(3) in your head. Answer: Q(2) can be easily calculated (mentally) than P(2) since in each step we have to perform linear operations starting from the inside. While evaluating P(2) mentally we find that larger number occurs in every step. To find P(2): 𝑃(𝑥) = 3𝑥 4 − 5𝑥 3 + 𝑥 2 − 3𝑥 + 5 𝑃(2) = 3(2)4 − 5(2)3 + (2)2 − 3(2) + 5 = 3(16) − 5(8) + 4 − 6 + 5 = 48 − 40 + 4 − 6 + 5 = 57 − 46 =11 To find Q(2): 𝑄(𝑥) = (((3𝑥 − 5)𝑥 + 1)𝑥 − 3) 𝑥 + 5 𝑄(2) = (((3(2) − 5)2 + 1)2 − 3) 2 + 5 =(((1)2+1)2-3)2+5 =((3)2-3)2+5 =(6-3)2+5 =3(2)+5 =6+5 =11 Now we have to write R(x) in nested form. 𝑅(𝑥) = 𝑥 5 − 2𝑥 4 + 3𝑥 3 − 2𝑥 2 + 3𝑥 + 4 Now we have factor out x. 𝑅(𝑥) = (𝑥 4 − 2𝑥 3 + 3𝑥 2 − 2𝑥 + 3)𝑥 + 4 = ((𝑥 3 − 2𝑥 2 + 3𝑥 − 2)𝑥 + 3)𝑥 + 4 = (((𝑥 2 − 2𝑥 + 3)𝑥 − 2)𝑥 + 3) 𝑥 + 4 = ((((𝑥 − 2)𝑥 + 3)𝑥 − 2) 𝑥 + 3) 𝑥 + 4 We can calculate R(3) mentally. R(x)=((((x-2)x+3)x-2)x+3)x+4 R(3)=((((3-2)3+3)3-2)3+3)3+4 =((((1)3+3)3-2)3+3)3+4 =(((6)3-2)3+3)3+4 =((18-2)3+3)3+4 =((16)3+3)3+4 =(48+3)3+4 =(51)3+4 =153+4 =157 R(3)=157 The polynomials P and Q are the same by expanding Q is proved. Question #5 To determine whether the given sequence is arithmetic progression or not: 3, 9, 15, 21, 27, ⋯ Select the correct answer below: a) The sequence is arithmetic with d=5 b) The sequence is arithmetic with d=6 c) The sequence is arithmetic with d=7 d) The sequence is arithmetic with d=9 Answer: Step 1 𝑎2 − 𝑎1 𝑎3 − 𝑎2 𝑎4 − 𝑎3 𝑎5 − 𝑎4 =9−3=6 = 15 − 9 = 6 = 21 − 15 = 6 = 27 − 21 = 6 Step 2 We can observe that the difference between successive terms is always constant. This is the common diference (d) of the AP. Therefore, the given sequence is arithmetic progression with d=6. The answer: (b) Question #6 Determine whether the given sequence could be geometric, arithmetic, or neither. If possible, identify the common ratio or difference. 9, 13, 17, 21, ⋯ a) arithmetic: d=4 b) arithmetic: r=4 1 c) arithmetic: 𝑟 = 4 d) neither Answer: Step 1 For a sequence to be arithmetic sequence difference between consecutive terms must be equal. Mathematically, 𝑑 = 𝑡2 − 𝑡1 = 𝑡3 − 𝑡2 Consider the given sequence: 9, 13, 17, 21, ⋯ Step 2 Now check for common difference, d=13-9=4 d=17-13=4 This proves that, 4 = 𝑡2 − 𝑡1 = 𝑡3 − 𝑡2 Therefore, it is a arithmetic progression with common difference 4. Question #7 Consider the following sequence: {8, − 24, 72, − 216, 648} It this an geometric sequence? a) The sequence is geometric. The common ratio is: r=? b) The sequence is arithmetic. c) The sequence is neither geometric nor arithmetic. Answer: Step 1 Given sequence, 8, − 24, 72, − 216, 648 The general form of geometric sequence is 𝑎, 𝑎𝑟, 𝑎𝑟 2 , 𝑎𝑟 3 , 𝑎𝑟 4 , ⋯ For the sequence, the ratio of the consecutive terms is constant is known as geometric sequence and the constant term is common ratio r. 𝑎 𝑟 = 𝑛+1 , 𝑛 ≥ 1 𝑎𝑛 Step 2 𝑎2 𝑎1 𝑎3 𝑎2 𝑎4 𝑎3 𝑎5 𝑎4 = = = = −24 8 72 = −3 = −3 −24 −216 72 648 −216 = −3 = −3 As we can see the ratio of the consecutive term is constant, thus the given sequence is geometric with common ratio -3 . Step 3 Therefore, The given sequence is geometric. The common ratio is: r=-3 Thus, 1𝑠𝑡 option is correct. Question #8 To calculate: Polinomial equation with real coefficients that has roots 1, 2 − 3𝑖. Answer: Formula Used: (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 Calculation: If the polynomial has real coefficients, then it's imaginary roots occur in conjugate pairs. So, a polynomial with the given root 2-3i must have another root as 2+3i Since each root of the equation corresponds to a factor of the polynomial, also, the roots indicate zeros of that polynomial, thus, the polynomial equation is written as, (𝑥 − 1)[𝑥 − (2 − 3𝑖)][𝑥 − (2 + 3𝑖)] = 0 (𝑥 − 1)(𝑥 − 2 + 3𝑖)(𝑥 − 2 − 3𝑖) = 0 Further use arithmetic rule. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 Here, a=x-2, b=3i Now, the polynomial equation is, (𝑥 − 1)[(𝑥 − 2)2 − (3𝑖)2 ] = 0 Use arithmetic rule. (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 And 𝑖 2 = −1 Now, the polynomial equation is, (𝑥 − 1)(𝑥 2 − 4𝑥 + 4 + 9) = 0 (𝑥 − 1)(𝑥 2 − 4𝑥 + 13) = 0 (𝑥 3 − 𝑥 2 − 4𝑥 2 + 4𝑥 + 13𝑥 − 13) = 0 𝑥 3 − 5𝑥 2 + 17𝑥 − 13 = 0 Hence, the polynomial equation of given roots 1, 2-3i is 𝑥 3 − 5𝑥 2 + 17𝑥 − 13 = 0 Question #9 Polynomial equation with real coefficients that has roots i, 1+ii, 1+i Answer: Formula Used: (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏2 Calculation: If the polynomial has real coefficients, then it's imaginary roots occur in conjugate pairs. So, a polynomial with the given root i, 1+ii, 1+1i must have other roots as -i, 1-i-i, 1-i. Since each root of the equation corresponds to a factor of the polynomial also the roots indicate zeros of that polynomial. Hence, below mentioned equation is written. (𝑥 − 𝑖)[𝑥 − (−1)][𝑥 − (1 + 𝑖)][𝑥 − (1 − 𝑖)] = 0 (𝑥 − 𝑖)(𝑥 + 𝑖)(𝑥 − 1 − 𝑖)(𝑥 − 1 + 𝑖) = 0 (𝑥 − 𝑖)[𝑥 − (−𝑖)][𝑥 − (1 + 𝑖)][𝑥 − (1 − 𝑖)] = 0(𝑥 − 𝑖)(𝑥 + 𝑖)(𝑥 − 1 − 𝑖)(𝑥 − 1 + 𝑖) = 0(𝑥 − 𝑖)(𝑥 + 𝑖)[(𝑥 − 1) − 𝑖][(𝑥 − 1) + 𝑖] = 0 (𝑥 − 𝑖)(𝑥 + 𝑖)[(𝑥 − 1) − 𝑖][(𝑥 − 1) + 𝑖] = 0 Further use arithmetic rule. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏2 Now, the polynomial equation is, (𝑥 2 − (𝑖)2 ][(𝑥 − 1)2 − (𝑖)2 ] = 0(𝑥2 − (𝑖)2)[(𝑥 − 1)2 − (𝑖)2] = 0 Use arithmetic rule. (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏2 And 𝑖 2 = −1𝑖2 = −1. Now, (𝑥 2 + 1)(𝑥 2 − 2𝑥 + 1 + 1) = 0 (𝑥 2 + 1)(𝑥 2 − 2𝑥 + 2) = 0 (𝑥2 + 1)(𝑥2 − 2𝑥 + 1 + 1) = 0(𝑥2 + 1)(𝑥2 − 2𝑥 + 2) = 0(𝑥4 + 𝑥2 − 2𝑥3 − 2𝑥 + 2𝑥2) = 0𝑥4 − 2𝑥3 − 2𝑥 + 2=0 (𝑥 4 + 𝑥 2 − 2𝑥 3 − 2𝑥 + 2𝑥 2 ) = 0 𝑥 4 − 2𝑥 3 + 3𝑥 2 − 2𝑥 + 2 = 0 Hence, the polynomial expression of given roots Hence, the polynomial equation of given roots i, 1+ii, 1+i is 𝑥 4 − 2𝑥 3 + 3𝑥 2 2𝑥 + 2 = 0𝑥4 − 2𝑥3 + 3𝑥2 − 2𝑥 + 2 = 0 Question #10 Consider the polynomial 2𝑥 3 − 3𝑥 2 + 𝑥 5 Which of the following statements about the polynomial are not correct? a) The standard form of the polynomial is 𝑥 5 + 2𝑥 3 − 3𝑥 2 . b) The polynomial is a trinomial. c) The polynomial is quintic. d) The polynomial has a linear term. Answer: Step 1 a) Standard form of polynomial is written with highest degree term as first term, second highest degree term as second term and so on. For the given polynomial, the standard form is: 𝑥 5 + 2𝑥 3 − 3𝑥 2 b) Trinomial is a type of polynomial which has 3 terms. The given polynomial is a trinomial as it has 3 terms, i.e., 𝑥 5 − 1𝑠𝑡 𝑡𝑒𝑟𝑚 2𝑥 3 − 2𝑛𝑑 𝑡𝑒𝑟𝑚 3𝑥 2 − 3𝑟𝑑 𝑡𝑒𝑟𝑚 Step 2 c) The polynomials having degree 5th are known as quintic polynomials. Since, the polynomial 𝑥 5 + 2𝑥 3 − 3𝑥 2 have degree =5 Hence, it is quintic polynomial. d) Linear term is term which have degree =1 Since, the polynomial 𝑥 5 + 2𝑥 3 − 3𝑥 2 has no term having 5th degree, hence the statement is not correct.