Question

Find the remainder:

1. P(x)=2x^100 divided by (x-1)

2. P(x)=3x^200-5x^100+1 is divided by (x+1)

3. Find the value k so that when P(x)=x^3+3x^2+kx-3 divided by (x-2) the remainder is -17.

note: I know how to use synthetic and long division on this but i just can't get the answer. Pls help me how to solve this.

Answer 1

you can use the divison algorithm to find the remainder, a = bq + r

so for number 1
\[2x^{100} = (x-1)q + r\]
let x = 1 and solve for r

Answer 2

division algorithm? okay wait im gonna try this

Answer 3

ok im confused help pls

Answer 4

divison algorithm is: Let a be an integer and b be a positive integer. Then there exist unique integers q and r such that a = bq + r, where 0 <= r < b

Answer 5

so for number 1 you plug in x = 1 you get

\[2(1)^{100} = (1-1)q + r\]
so you get 2(1) = 0q + r
2 = 0 + r
r = 2
so your remainder is 2

Answer 6

do you understand what i did?

Answer 7

why is that q was 0?

Answer 8

0 multiplied by anything is 0

Answer 9

i let x = 1 since i know 1-1 = 0 and 0q = 0 so i can get r by itself

Answer 10

so q is equals to 0?

Answer 11

ohh i get it.. !

Answer 12

you pretty much want to make q equal to 0 so you can solve for r

Answer 13

okay one more question in question #3 how can you find the value of k?

Answer 14

you can use the division algorithm again

\[x^3 + 3x^2 + kx - 3 = (x-2)q - 17\]
let x = 2 and get:

\[2^3 + 3(2^2) + 2k - 3 = (2-2)q - 17 \]
so you get 8 + 3(4) + 2k - 3 = 0 -17
8 + 12 + 2k - 3 = -17
20 + 2k - 3 = -17
17 + 2k = -17
2k = 0
k = 0

Answer 15

k = -17

Answer 16

In general, if a polynomial f(x) is divided by (x - a), then the remainder is f(a).

Answer 17

hm.. im off by a sign

Answer 18

but k = - 17 is correct

Answer 19

o wait i see i made a stupid algebraic mistake

17 + 2k = -17
2k = -34
k = -17 sorry haha

Answer 20

happens to all of us :)

Answer 21

For part 2) Find remainder when P(x)=3x^200-5x^100+1 is divided by (x+1)
(x+1) = (x -(-1)) so a = -1.
P(-1) = 3(-1)^200 - 5(-1)^100 + 1 = ? that will be your remainder.

Answer 22

Thank u !! I can finally answer this other questions!!

Answer 23

yw.

Answer 24

your welcome glad to help