Question

Find the remainder when x100 is divided by x2-3x2

Answer 1

HEY!!

We know that Divided = Quotient × Divisor + Remainder

⇒ Quotient × Divisor = Divided – Remainder

i. e. Division is a factor of (Dividend – Remainder)

⇒ (x 2 – 3x + 2) is a factor of (x 100 – Remainder)

⇒ (x – 1) and (x – 2) are factor of (x 100 – Remainder)

[ x 2 – 3x + 2 = x2 – 2x – x + 2 = x (x – 2) –1 (x – 2) = (x – 1) (x – 2)]



Let f (x) = x 100 – Remainder

⇒ f (1) and f (2) = 0



∴ Remainder = 2 100 ( x – 1) – ( x – 2)

[ f(1) = 1100 – [2100 (1 – 1) – (1 – 2)] = 1 – [0 + 1] = 1 – 1 = 0

and f(2) = 2100 – [2100 (2 – 1) – (2 – 2)] = 2100 – (2100 – 0)= 2100 – 2100 = 0]

Answer 2

0 is remainder as x^100 is divisible by -2x^2