math112

MATH 112 Fall 2018 - GHW 3 P3 Name _________________________________________________ This GHW is due at the beginning of class on Friday, October 26. You may discuss this assignment with others in the class or with me, but the work that you turn in must be your own write-up. For each of the following problems, supporting work is necessary for credit (unless stated otherwise). Supporting work must be presented in a complete, neat and orderly fashion. This includes the use of correct notation. Put your work and answers (unless stated otherwise) on one side of separate standard size sheets of paper. Your work must be stapled (in the upper left-hand corner) to this sheet with this sheet being the first page, and with the problems given in order. Be sure to justify and simplify your answers. Remember to use pencil. Express all numerical answers using exact values.

1. For each of the following polynomials, using the Rational Zeros Theorem, determine a list of all possible rational

zeros. Make the list as short as possible using the information that is given. Do not attempt to find the actual zeros.

a) 7 6 4( ) 6 15 3 21 24g x x x kx x     where k is an unknown integer.

b) 4 3 24( ) 3 2 4

5 h x x x x x     .

2. Let 6 5 4 3( ) 4 7 11 6 16 12f x x x x x x     

a) Verify that 3

4 is a zero of f using synthetic division.

b) Verify that 1 i is a zero of f using synthetic division.

c) Verify that 1 i is a zero of f by doing the following:

i. Find real numbers b and c so that 1 i is a zero of 2

x bx c  .

ii. Show that 2

x bx c  as found in the previous part is a factor of f.

d) Completely factor f as a product of linear factors.

e) Completely factor f over the real numbers, that is, express f as a product of linear and irreducible quadratic

factors with real coefficients.

3. Find the polynomial f of lowest degree with real coefficients such that 1 and 3 i are zeros of f having multiplicities 2 and 1 respectively, and such that (3) 20f  . You only need to expand f until all complex numbers are removed.

4. For each of the following, determine the minimum degree a polynomial f must have if it satisfies the given conditions. Briefly justify your answer.

a) ( 3) 0f   , ( 1) 0f   , (4) 0f  and f changes sign at 1x  and at 3x  but not at 4x  .

b) ( 3) 0f   , ( 1) 2f   , (1) 5f  , (3) 0f  and (4) 0f  .