Question

A quadratic polynomial when divided by x-1 leaves a remainder of 1 and when divided by x+1 leaves a remainder of -3

Answer 1

Solution for this problem is only possible when the leading coefficient of the polynomial is 1.

Let the polynomial be p(x)
Since p(x) is a quadratic polynomial
Therefore it can be written as
P(x) =ax²+bx+c
Now, considering the question and using remainder theorem, we get
p(1) =1
=>a(1)²+b(1)+c=1
=>a+b+c=1 ..............................(1)
Also,
p(-1) =-3
=>a(-1)²+b(-1)+c=-3
=>a-b+c =-3 ...............................(2)
On subtracting 1 from 2, we get
a-b+c-a-b-c=-3-1
=>-2b=-4
=>b=2

Now we have
a=1 ( as mentioned above the leading coefficient of the polyomial is 1)
And,
b=2
On putting these values in 1 we get
1+2+c=1
=>c=-2
Hence the required polynomial is
x²+2x-2

You can also check if my solution is right or not
Here,
p(x) =x²+2x-2
and when it is divided by x-1 the remainder can be found using the remainder theorem. i.e.
P(1)= 1+2-2=1
Similarly
P(-1)=1-2-2=-3


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Regards,
Gaurav kumar