Question

Find the remainder when f(x) is divided by g(x)=x+1
f(x)=8x^63-12x^57+16x^48+5x^23-7x^13-17

Answer 1

use the remainder theorem

Answer 2

if
\[\large f(x)=(x-a)\cdot q(x)+r(x) \]
then
\[\large f(a)=(a-a)\cdot g(a)+r(a)=r(a) \]

Answer 3

okay mind talking me through it as I set up the problem?

Answer 4

plugin x = -1 in f(x)

Answer 5

f(x)=8x^63-12x^57+16x^48+5x^23-7x^13-17

f(-1) = ?

Answer 6

So just sub in (-1) every place there is an x?

Answer 7

yes!

Answer 8

5?

Answer 9

\[8(-1)^{63}-12(-1)^{57}+16(-1)^{48}+5(-1)^{23}-7(-1)^{13}-17\]

Answer 10

f(x)=8x^63-12x^57+16x^48+5x^23-7x^13-17

f(-1) = -8 + 12 + 16 -5 + 7 - 17
= 5

Answer 11

good job !

Answer 12

in simple terms,
remainder theorem says this : the remainder when u divide any polynomial \(f(x)\) by a linear polynomial \((x-\color{red}{a})\) is same as \(f(\color{red}{a})\)

Answer 13

Ill have to go through my notes again for that one. Seemed so simple it confused me lol. So my answer would just be 5? Even though g(x)=x+1?

Answer 14

yes, 5 is the remainder

Answer 15

So it could be set up how we did earlier with long division, but this way is just A LOT easier? or no?

Answer 16

if u setup long division,
you wont be able to finish it in ur life time -.-
try it :)

Answer 17

but when u finish, you wud get the same 5 as remainder

Answer 18

Hahaha okay I was just making sure. God that would be cruel punishment for any man (and woman :p)

Answer 19

What about proving roots?

Answer 20

lol yes, nevertheless its a good idea to try long division and see why its really called "long division"

Answer 21

Lmao

Answer 22

you will get 63 rows in ur long division by the time u finish and see the remainder

Answer 23

http://www.wolframalpha.com/input/?i=long+division+%288x%5E63-12x%5E57%2B16x%5E48%2B5x%5E23-7x%5E13-17%29%2F%28x%2B1%29

Answer 24

Knowing me I would miss a negative and completely ruin the problem .-.

Answer 25

long division is insanely long for this particular problem,
try it and see lol

Answer 26

no no I trust you lol. Now for trying to prove roots I cannot find anything in my notes and I like to be "ahead of the game" I guess you can say so mind helping with a practice problem from book?

Answer 27

Or I can open new question

Answer 28

shoot..

Answer 29

Prove g(x) has a root between (1,2)

\[g(x)=x^5-4x^4+7x^3-2x^2+x-5\]

Answer 30

intermediate value theorem

Answer 31

let me ask u a simple q :-
wat does it mean to have a root between (1, 2) ?

Answer 32

having a root means between (1, 2) means this :-
the function must equal "0" for some number between (1, 2)

Answer 33

a graph might help