Question

Name Math 1314 Lab Module 6 For each of the following problems, show all work. Simplify and clearly indicate all answers. 1. For the function f(x) = 3x - 4x° + x° +6x-2 a. State the degree of the polynomial. b. State the number of zeros the polynomial function will have. C. Use the Rational Zero Theorem to list all of the possible rational zeros d. Use trial and error to find two zeros of f(x) then write them in the table below. Zeros Due: Upload to Lab 6 Canvas Dropbox by Sunday, August 2 by 11:59

only questions "d through f"

Answer 1

f(3) = 37"-4** +x?+62-2 d.) f(-1) = 3609-44-0 + (-1) +6-(-1)-2 = 3 +4+1-6-2 - 8-8 =0 Thus X-1 is a root of f(x). Then we try fo) =-2 +0 f(1) = 3 - 4+1+6-2 =4 +0 f(3) = 48 - 32 +4 +12-2 30 70 Now at 23-2, f(-2) = 36-294-46-2+(-2)761-3) -2 =48 +32+4-12-2 70 70 x = 4 f(4) = 2/10 at + -2 8 *+- Lo +5 - į +3-2 16 2 1 6 + 4 + >0.

and f (42) yo 1 we see that f (0) <0 so there must be a root between 0 and į: at x = și 5(!)- 3.(1)4 4(!) (1) 76(4)-2 15 + 2-2 1/4 + 6 3: 81 4 27 + 27 |-4+3 27 =0 lie, f()=0. Hence, X Ś is a root of f(x). Zeros x= x= -1 š as .(e.) Now we find the final two heros. x= -1 and x = are roots of f(x), SO (+1) (8-) must be a factor [(x+1)(2-1) it 2 x - 11/13 Them f(x) = (2+1) (x-1) g(x) where g(x) is a quadratic polynomial given by f (2) = (3x-x+6) (2+1) (2=1/) of f(x). g (x)

Thus trany J) = (x+1) (x - ) (32=6x+6) ->(*) = 3 (2+1) (X - $) (**–2x+2) We have to find the roots of the quadratic 22 20th whose noots polynomial are given by 2. I 1-272-4.2.1 2= 2.1 2 I J 4 - 8 - 2 2 + 2i 2. 1ti 2 Thus the final roots are Hi and t-i Zeros ti 1-i (fi) since, all the heros of f(n) are -4, §;iti, isi f(x) as then we can write 1 $(2) = (x+1) (x - $) {x-(1+i)} (-(1-1)}