functions + remainders question!

Question

jadedry · Answer

$$f(x) = ax^2 +bx +c $$

leaves remainders 1,25,1 on division by $$x-1$$
$$x+1$$

$$x-2$$

respectively. show that $$f(x)$$ is a perfect square.

jadedry · Answer

I did a bit of working, but it doesn't really go anywhere + is probably incorrect. Can anyone give me the first step or so?

Thanks in advance!

jadedry · Answer

Yes, I do. c: 

essentially if,  a f(x) is divided by (bx-A) , the remainder is (f)A/b, right?

jadedry · Answer

You have to plug in A/b into (f)x equation? (replace x with A/b)? 

I'm not entirely sure of your question? :c

jadedry · Answer

So: if x = -1

$$f(-1) = a - b +c $$ R1

therefore the equation is really:

$$a -b +(c -1)$$

right?

jadedry · Answer

wait, not minus b:
x-1 =

$$a+b+(c+1)$$
----

x + 1

=

$$a - b +c +25$$

jadedry · Answer

x-2 =

$$4a +2b +(c+1)$$

jadedry · Answer

at this point, I'm really not sure how to go on and prove it is a square...

imqwerty · Answer

the remained that u get on dividing a polynomial lets say $$p(x)=x^3-2x^2+x+1$$  with some number say  x-3
this is just an example

ok so the remainder on dividing it with x-3 will be equal to the value of p(x) when u put x=3

if we were to divide it by 2x+1  and then we wanted to find the remainder
then the remainder wuld be the value of p(-1/2)

can u apply this logic...u'll get 3 equations and then find a,b,c 

after finding a,b,c from them 
put the values of a,b,c in the given equation
u'll see that it is of the form of a perfect square

jadedry · Answer

@imqwerty 

So I just have to find the values of a, b and c then? no abstract concepts?

imqwerty · Answer

yea  (:
just find a,b,c then u'll understand

jadedry · Answer

Okay, sounds simple enough haha.

Just to confirm, could you please check my equations for x-1, x+1 and x-2 (just a bit up) and see if they're correct so far?

imqwerty · Answer

hmm i didn't understand it 

it is like this--->

1st equation when we divide by x-1

so the remained will be the value of ax^2 +bx+c when x=1
so
remainder =a(-1)^2 +b(-1) +c
we know that remainder=1
so
1=a-b+c
 
make equations like this

jadedry · Answer

Ah yes, I understand, that would be easier to work with, I'll keep that in mind.

In the end I have managed to come up with a perfect square equation, so your advice has definitely helped. (hate it when it's something super simple that trips me up, ugh)

Thank you! c:

imqwerty · Answer

yw :D  jst keep basic concepts in mind they help u to understand and solve difficult prblms (: