MA134 Sample Problems for Exam #2 Part I. 1. What is the degree and the leading coefficient of the polynomial: x5 + √ 12 84x6 + πx3 − 4.1x14 + 8x − 13x10 + x13 − 10 + 0.1x8 . 7 degree: leading coefficient: 2. The graph of a polynomial P (x) is given below: y 6 y = P (x) 1 -2 0 1 3 What are the roots of P (x)? -x How small the degree of P (x) can be? What is the sign of the leading coefficient? Use the graph to solve: P (x) < 0. 3. Find the quotient and remainder x4 − x3 + x2 − x + 2 . x−2 quotient: remainder: 4. Find the remainder of the division 6x2001 − 17x1000 + 12x100 + 26 by x + 1. remainder: 5. Write an example of a polynomial which has no real roots. 6. Polynomial p(x) has real coefficients and degree 4. Three of its zeros are 1,2,1 + i. Find the fourth zero of p(x). 1 7. Compute i2001 , |3 − 2i|, 5 − 11i. √ 8. Evaluate log2 2 3 , log3 9π , 5log5 2001 , 7log49 2001 . 9. What are the coordinates of the vertex of the parabola y = (x + 1)2 − 4, y = −4x2 + 4x? 10. Find the domain of each functions f (x) = log2 x2 , g(x) = log(x2 − 3x + 2), h(x) = log x+1 x . Part II. 11. Solve: 2x4 + 3x3 − 4x2 − 3x + 2 = 0. 12. Polynomial S(x) = ax8 − bx2 − cx + d has integer coefficients. Three roots of S(x) are known: −2/3, 1/4 and 7. Coefficients a and d are between 25 and 40. Find a and d. 13. A rectangular parcel of land has an area of 1000 ft2 . A diagonal between opposite corners is measured to be 10 ft longer than one side of the parcel. What are the dimensions of the land, correct to the nearest foot? P (x) 14. Construct a rational function Q(x) such that it has vertical asymptotes x = 1 and x = −4, horizontal asymptote y = −15 and x-intercepts 2 and 3. 15. Solve log6 (x + 3) + log6 (x + 4) = 1. 3 2 16. Solve 2x = 3x . 17. How long does it takefor an investment to triple in value if it is invested at 10% per annum compounded momnthly? 18. After the explosion and release of radioactive material into the atmosthere from the nuclear power plant at Chernobyl (former USSR, now Ukraine) in 1986, the hay in Austria was contaminated by iodine-131 (half-life 8 years). If it is all right to feed the hay to cows when 10% of the iodine-131 remains, how long do the farmers need to wait to use this hay? 19. Function f is one-to-one. Find the inverse of f and check your answer. 20. A landscape engineer has 200 feet of border to enclose a rectangular pond. What dimensions will result in the largest pond. Extra credit, 10 points. P (x) in Problem 2 has the lowest possible degree and the y-intercept is −1. Find P (x). 2