Find remainder

Algebra Level 4

What is the remainder when $x^{2016}$ is divided by $(x-1)(x-2)$ ?

(2^{2016})x +2-2^{2014}

(2^{2016}-3)x -2+2^{2016}

(2^{2016}-1)x +2-2^{2016}

0

2 solutions

D H
Aug 25, 2016

On dividing (x^{2016}) by (x-1)(x-2) remainder will be ax + b Then (x^{2016}) + ax + b is exactly divisible by (x-1)(x-2)

.'. (1^{2016}) + a + b =0 => a+b = -1

and also , (2^{2016}) + 2a + b = 0

=> (2^{2016}) + a +(-1) = 0 => a =1 - (2^{2016}) => b = (2^{2016}) - 2

.'. (1 - (2^{2016}) ) x + (2^{2016}) - 2 is a factor of (2^{2016})
=> ( (2^{2016}) - 1 ) x + 2 - (2^{2016}) is a factor of (2^{2016})

Sundara Karthik
Aug 24, 2016

whenever any polynomial is divided by quadratic then the remainder will be of form ax+b...

Acc. to remainder theorem..

f(x)=g(x)*q(x)+r(x), f(x) is a function, g(x) is a function which is dividing f(x), q(x) is the quotient when f(x) is divided by g(x), r(x) is the remainder.

=> x^2016=q(x)*(x-1)(x-2)+ax+b

put x=1, x=2 you will two equations in a and b solve them and you will arrive at the answer......

Find remainder

2 solutions

0 pending reports