Question

Consider the Polynomial \(P(x)\). The remainders of \(P(x) \div (x-2)\) is -10 and \(P(x) \div (x+2)\) is \(4\) respectively. Find the remainder of the division \(P(x) \div (x^{2}-4\)

Answer 1

Well, how do i start? I haven't done these in a while

Answer 2

for the first division
x-2 = 0
x=2
therefore P(2)=-10

Answer 3

for the second division
x+2=0
x=-2
therefore P(-2)=4

Answer 4

for the last division
which is ur question actually
x^2-4=0
x = +2 or -2
two cases therefore
u must take P(+2) & P(-2) both
& u know the values of these

Answer 5

the values are remainders actually
given by Remainder Theorem.

Answer 6

I think I got it. I assume there's a formula for the remainder?

Answer 7

wait lemme google.

Answer 8

ok:)

Answer 9

So it's \(P(x)=(x-a)q(x)+r(x)\)
then \( P(x)-(x-a)q(x) = r(x)\)?

Answer 10

yeah

Answer 11

what is \(q(x)\)?

Answer 12

that is quotient function
but i think
it is looking more like division lemma

Answer 13

Actual Remainder theorem : -

"If f(x) is a function; & (x-a) is its factor then f(a) = 0."
If f(x) is a function; & (x-a) is its factor
This statement means that
f(x) is divisible by (x-a)

Answer 14

how am i able to find the remainder? through assumptions?

Answer 15

i have given that
if (x-a) is factor of f(x) then; the remainder will be zero becoz f(a)=0
but if suppose (x-a) is not its factor then the remainder will be f(a) = something but not zero.

Answer 16

which means its \(0\)..?

Answer 17

sorry!

Answer 18

wt do u mean?

Answer 19

i mean wt do u wanna ask actually?

Answer 20

Well, from what you stated. You are saying that the remainder is \(0\)? Which I highly doubt is not.

Answer 21

The remainder is zero when & only when x-a is factor of f(x) or in other words f(a) = 0 otherwise not.

for example take ur question
for the first division u take
x-2 = 0
x=2
but since remainder is -10
then it means that x-2 isn't a factor of P(x)
& one more thing P(2) is nt equal to zero since x-2 isn't a factor of P(x)
but we know remainder is -10
therefore P(2) =-10

Answer 22

go through my statements clearly
u would understand it much better.

Answer 23

alright. thanks

Answer 24

welcome:)

but do more questions based on remainder theorem
gradually
u would understand it:)