MT1710: Question Sheet 2 To be handed in: 16 October, 2018 (by 1pm, in the Maths office, McCrea Room 118). Questions 6 to 10 are in the other side of this sheet and are part of this Question Sheet. For that you'll need to use Mathematica. It is compulsory to have done a good attempt at those questions to gain the mark towards your continuous assessment. With your solutions you need to give in a print out of the Mathematica's solutions you complete! You can't get a mark '1' without this part com- pleted. 1. Find the real roots (solutions) of the following equations by factorizing the polyno- mials: (a) x2+ 2x- 3 =0, (b) x2 + 4x - 1 = 0, (c) x3+ x2-7x+5 =0, (d) x3+1 =0, (e) x3- 7x+6 =0, (f) x4- 4x3 - x2 + 16x-12=0. 2. Find a quartic equation which has as its solutions x =- 1, x=2, x = - 1+13 , 2 - 2 1- v3 . 3. Divide the polynomial x5 - 8x3 + 2x2 + 1 by x2 - 2x + 2 to find the quotient and the remainder. 4. Given that the remainder when x4 - ax3 + 2x + 1 is divided by x - 1 is the same as the remainder when x3 + ax2 - ax + 2 is divided by x + 1, what is the value of a? 5. Sketch the graphs of y = x3 and y = x - 1 on the same diagram. Deduce that the equation x3 = x - 1 has one real root. Is this root positive or negative? How many real roots does the equation x4 = x2 - 1 have? Justify your answer. P.T.O.
The questions on this side of the sheet should be done using Mathematica. You need to print out the work you complete and it has to be handed in with the answers to Q.1-5. 6. Use Mathematica to check your sketches in your answer to Question Sheet 1, question 1. [You may need to combine Plots using the Which command or Show command to answer parts (a) and (b); there are more elegant ways, involving the use of a single Plot command. The command Abs[x] is Mathematica's answer to |x|.] 7. Consider the polynomial p(x)= 4x3 + 12x2 - x-15. Use "Factor" to factorize this polynomial and "Solve" to find the roots of the equation p(x) = 0. 8. The instructions "PolynomialQuotient[p(x), q(x), x]" and "PolynomialRemain- der[p(x), q(x), x]" divide the polynomial p(x) by the polynomial q(x), to give the quotient and remainder respectively. Use these instructions to check your answer to question 3 of this sheet. 9. Use "Solve" to find the exact values of the solutions of the equation x3 = x - 1 and "NSolve" to find the numerical values, correct to two decimal places. Relate this to your answer to Question 5 of this sheet. 10. Use Mathematica to solve the cubic x3 + y3 = 8xy ,