1.

0.Find the remainder when x 100 is divided by x? - 3x + 2.

Answer»

Letp(x)=x100p(x)=x100andq(x)=x2–3x+2=x2–2x–x+2=x(x–2)–1(x–2)=(x–2)(x–1)q(x)=x2–3x+2=x2–2x–x+2=x(x–2)–1(x–2)=(x–2)(x–1)

LetQ(x)Q(x)andr(x)r(x)be the quotient and remainder whenp(x)p(x)is divided byq(x)q(x). Hence by division algorithm, we havep(x)=q(x)⋅Q(x)+r(x)p(x)=q(x)⋅Q(x)+r(x)

That is,x100=(x–2)(x–1)Q(x)+r(x)x100=(x–2)(x–1)Q(x)+r(x)

Letr(x)=(ax+b)r(x)=(ax+b)where0<degr(x)<20<deg⁡r(x)<2

x100=(x–2)(x–1)Q(x)+(ax+b)(1)x100=(x–2)(x–1)Q(x)+(ax+b)(1)

Putx=1x=1in Equation(1)(1), we get1=0+(a+b)1=0+(a+b)

Hence(a+b)=1(2)(a+b)=1(2)

Now put x = 2 in equation (1), we get

2100=0+(2a+b)2100=0+(2a+b)

2a+b=2100(3)2a+b=2100(3)

Solving(2)(2)and(3)(3), we geta=(2100–1)a=(2100–1)andb=2–2100b=2–2100

Therefore the remainder=(ax+b)=(2100–1)x+(2–2100)


Discussion

No Comment Found

Related InterviewSolutions