Tomato Objective 288 | Finding big remainder in a small way

Try this problem from TOMATO Objective 288, useful for ISI BStat, BMath Entrance Exam based on finding big remainder in a small way.

Problem: Tomato objective 288

The remainder R(x) obtained by dividing the polynomial [latex]x^{100}[/latex] by the polynomial [latex]x^2-3x+2[/latex] is

(A) [latex]2^{100}-1[/latex]

(B) [latex](2^{100}-1)x-(2^{99}-1)[/latex]

(C) [latex]2^{100}x-3(2^{100})[/latex]

(D) [latex](2^{100}-1)x+(2^{99}-1)[/latex]

SOLUTION:  (B)

The the divisor is a quadratic term .So, R(x) must be 1 degree less than divisor.

We know , [latex]f(x)=divisor.Q(x)+R(x)[/latex]

[latex]=> x^{100}=(x^2-3x+2).Q(x)+ (ax+b)[/latex]

[latex]=> x^{100}=(x-1)(x-2).Q(x)+ (ax+b)[/latex]

when ,[latex]x=2[/latex]

[latex]=> 2^{100}=(2-1)(2-2).Q(x)+ (2a+b)[/latex]

[latex]=> 2^{100}= (2a+b)..........(i)[/latex]

when ,[latex]x=1[/latex]

[latex]=> 1^{100}=(1-1)(1-2).Q(x)+ (a+b)[/latex]

[latex]=> 1^{100}=(a+b)...........(ii)[/latex]

solving two equation we get, [latex]a=(2^{100}-1)[/latex]

and, [latex]b=-(2^{99}-1)[/latex]

The remainder R(x) is [latex](2^{100}-1)x-(2^{99}-1)[/latex]

Some Useful Links:

• ISI BStat Entrance - Solutions and Problems
• ISI CMI Entrance - BStat Program
• Solutions of Equations, ISI 2018 Problem 1 - video