Question

Help Please. I'm going out of my mind. I thought this seemed pretty straight forward, but I have tried it literally 17 times and I cannot for the life of me figure it out. help me please. here is the problem:

f(x) = x^2+2x+3 and g(x) = 4x^3-5x^2+4x+1

For brevity let's write deg (p) for the degree of a polynomial p. So in the above example, deg (f) = 2 and deg (g) = 3. You can answer the following questions without actually computing the indicated functions.
deg (f+g) .
deg (f-g) .
deg (fg) .
deg (f o g) .
deg (g o f).

Answer 1

deg (f+g) = max deg (f,g)
deg(f-g) = max deg (f,g)
deg (f*g) = deg(f) + deg (g)

Answer 2

deg (f+g) is 3
deg (f-g) . is 3
deg (fg) . is 6

Answer 3

oops deg(fg) is 5

Answer 4

deg of (fog) is 6
deg (g o f) is 6

Answer 5

deg (f +/- g) = max {deg (f), deg(g)}

Answer 6

how are you getting that without having to compute?

Answer 7

you can ignore the other terms, the dominant terms are the highest degree

Answer 8

ohh! I think i was over thinking it and over computing. thank you so much!

Answer 9

follow the rules as defined by
perl

Answer 10

@perl
i wrote for f o g or gof but next i hv corrected it

Answer 11

deg (f+g) = max {deg (f), deg(g)}
deg(f-g) = max {deg (f) deg,(g)}
deg (f*g) = deg(f) + deg (g)
deg ( f o g ) = deg (f) * deg(g)
deg (gof) = deg (f) * deg (g)

we should prove this, but shouldnt be hard
basically i got the formulas from just manipulating the dominant terms

Answer 12

@matricked
that was correct, the deg(f*g) was 5, but you corrected

Answer 13

for f+g the dominate degrees are 2 and 3, wouldn't that make the answer 5 and not 3?

Answer 14

i have to make some conditions on my rules
Theorem 1.1 Let R be a ring and f, g ∈ R[x] {0}. The following statements are true.

Answer 15

(i) fg = 0 or deg (fg) ≤ deg f + deg g.
(ii) f + g = 0 or deg (f + g) ≤ max{deg f, deg g}.
(iii) If deg f = deg g, then deg(f + g) = max{deg f, deg g}.
(iv) fg = 0 or ord (fg) ≥ ord f + ord g.
(v) f + g = 0 or ord (f + g) ≥ min{ord f, ord g}.
(vi) If ord f = ord g, then ord(f + g) = min{ord f, ord g}.

Answer 16

wow. I'm trying to study this for my exam in about 15 days but i fell confidant that I will bomb any problem like this

Answer 17

I would just manipulate the dominant terms
,when you multiply the two expression, the other terms are all smaller

Answer 18

( x^2+2x+3)
(4x^3-5x^2+4x+1) , for example if you multiply them you get
------------------------
4x^5 + smaller power terms

Answer 19

so the degree of the product of the polynomials is going to be 5

Answer 20

oh, I think I see what you are saying.

Answer 21

assuming you put the polynomials in descending order (standard form)

Answer 22

the only tricky one is the composition

Answer 23

and maybe division, but you werent asked

Answer 24

Thank you. I'll probably be studying this and trying to understand it, still feels foreign.

Answer 25

:)

Answer 26

try the composition , using the idea that you only need the dominant terms

Answer 27

fog = f(g(x)) = f ( 4x^3 + smaller terms ) = (4x^3 + smaller terms ) ^2
= (4x^3) ^2 + smaller terms
= 16x^6 + smaller terms, and the degree is 6

Answer 28

ohmigawd....i think i really understood what you just wrote! Thank you jezus! seriously, thank you for your help!

Answer 29

:)

Answer 30

finding general rules can be tricky, since there are different cases to consider. so you can ignore the rules i wrote above, i have to double check them

Answer 31

okay

Answer 32

and just use the trick of looking at the dominant terms plus smaller terms

Answer 33

I will thank you.