Name: Math 114 Activity 1(Due by EOC Apr. 17) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please allow them to work the problems on their own! If you would like to help them with similar problems, here are the related homework problems: pg. 544:17-71 odd, 81-95 odd. Thanks! Dear Student, Don t even think about asking for help on the problems in this activity! The recommended homework problems have answers in the book, so do them first for practice!!! Use the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-) 4 f x x x 6x 8. f x x x 11x 9x 15 1. List all the possible rational zeros. Use synthetic division to find all the zeros of the following polynomials. (-6). f x x x 11x 1 4. f x x x x 1 Possible rational zeros: Possible rational zeros:
4 5. f x x 5x 17x 1 6. Possible rational zeros: f x x x 16x 15 Possible rational zeros: Find an n th -degree polynomial function with real coefficients satisfying the given conditions. (7-10) 7. f 1 50 8. n ; 6 and 5 i f 66 n ; 4 and i are zeros; are zeros;
1 9. n 4;,, and i are zeros; f 1 18 10. n 4; 4, 1, and i are zeros; f 1 18 Use Descarte s Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomials. (11-1) f x x 7x x 7 1. f x x x x 7 11. 1. 4 f x 4x x 5x x 6
Find all the zeros of the following polynomial functions. (14-16) f x x 1x 1x 10 14. Possible rational zeros
15. 4 f x x 4x x 14x 10 Possible rational zeros
16. 5 4 f x 4x 1x 41x 99x 10x 4 Possible rational zeros
17. Find the exact number of positive and negative zeros of the polynomial function f x x 5x 1. 4 18. Find a, b, and c for f x ax bx c if f 4i 0, f 4i 0, and f 1 0. 19. Find values of a and b so that the polynomial ax b x 5 1 5x 1 x 1. is divisible by
0. There is an imaginary version of the Rational Root Theorem for finding roots of polynomials: n n 1 If P x anx an 1x a1x a0 is an n th degree polynomial with integer coefficients, and x i is a rational imaginary zero of P x, then r must be a factor p r q r of a n and p q must be a factor of a 0. 4 a) For P x x x 8x 6x 15, let s look at the possibilities: Possible Rational Imaginary Zero Possible Factor 1 i x x 5 1 i x x 5 Test the possible rational imaginary zeros, and then find all the zeros. 4 b) For P x x 4, let s look at the possibilities: Possible Rational Imaginary Zero Possible Factor i x 1 1 i x x 1 i x x i x 4 Test the possible rational imaginary zeros, and then find all the zeros.
1. There is a cubic formula similar to, but more complicated than the quadratic formula. Let s derive it in the special case of cubic equations of the form x px q 0. First, make the substitution x u v to get u v p u v q 0 u u v uv v pu pv q 0 u v uv p u v q 0 We can solve the cubic if we can find u and v that satisfy u v q p. uv p 6 p This system is equivalent to u q u qu 0. This is a quadratic u 7 equation in u q q p u, so from the quadratic formula, we get. To get v, use the fact that 4 7 So solutions of x x px q 0 are given by u p v, to get v u 4 p q q 7 or p. q q p 4 7 q q p p. Since the cube roots can produce as many as 4 7 q q p 4 7 numbers, it looks like you could get a lot of solutions, but you will get at most. Since the equation is a cubic, once you get one solution, the equation can be reduced to a quadratic. a) Use the formula to find one solution of the cubic equation x 6x 9 0. b) Use the formula to find one solution of the cubic equation x 6x 8 0.
Bonus#1:
Bonus#: What fractional part of the figure is shaded(assuming that line segments that appear parallel actually are, all angles are right angles, and the vertical line segments are equally spaced.)? Bonus#: Find two different numbers so that each of them is the square of the other number.