Question

The remainder when x 1999 is divided by x 2 – 1 is (1) – x (2) 3x (3) x (4) None of these

Answer 1

p(x)= x^1999
factors of x^2-1=(x+1)(x-1)
we know,
dividend=divisor*quotient+remainder
since p(x) is a polynomial of degree 1999 so it will leave a linear remainder in form of ax+b where a and b are constants.
x^1999=divisor*(x+1)(x-1)+ax+b
put x=1
so, 1=a+b                             ..................(i)
put x=-1
so, -1=-a+b                       .................(ii)
solving equations
we get,
a=1 and b=0
so remainder=ax+b=1*x+0=x
so when x^1999 is divided by x^2-1 then it leaves remainder as x.


option (3) is correct.

Answer 2

Answer:

p(x)= x^1999

factors of x^2-1=(x+1)(x-1)

we know,

dividend=divisor*quotient+remainder

since p(x) is a polynomial of degree 1999 so it will leave a linear remainder in form of ax+b where a and b are constants.

x^1999=divisor*(x+1)(x-1)+ax+b

put x=1

so, 1=a+b                             ...(i)

put x=-1

so, -1=-a+b                       ..(ii)

solving equations

we get,

a=1 and b=0

so remainder=ax+b=1*x+0=x

so when x^1999 is divided by x^2-1 then it leaves remainder as x.

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