1 option is the right answer

first option is rigjt

The given polynomial isx3+3x2+3x+1x3+3x2+3x+1.

By remainder theorem, ifp(x)=x3+3x2+3x+1p(x)=x3+3x2+3x+1is divided byx+1x+1then remainder is determined byp(−1)p(−1).

p(−1)=(−1)3+3(−1)2+3(−1)+1=−1+3−3+1=0p(−1)=(−1)3+3(−1)2+3(−1)+1=−1+3−3+1=0

Thus, the remainder is 0.

iiii

The given polynomial isx3+3x2+3x+1x3+3x2+3x+1.

By remainder theorem, ifp(x)=x3+3x2+3x+1p(x)=x3+3x2+3x+1is divided byx−12x−12then remainder is determined byp(12)p(12).

p(12)=(12)3+3(12)2+3(12)+1=18+3×14+3×12+1=278p(12)=(12)3+3(12)2+3(12)+1=18+3×14+3×12+1=278

Thus, the remainder is278278.

iiiiii

The given polynomial isx3+3x2+3x+1x3+3x2+3x+1.

By remainder theorem, ifp(x)=x3+3x2+3x+1p(x)=x3+3x2+3x+1is divided byxxthen remainder is determined byp(0)p(0).

p(0)=(0)3+3(0)2+3(0)+1=0+1=1p(0)=(0)3+3(0)2+3(0)+1=0+1=1

Thus, the remainder is 1.

iviv

The given polynomial isx3+3x2+3x+1x3+3x2+3x+1.

By remainder theorem, ifp(x)=x3+3x2+3x+1p(x)=x3+3x2+3x+1is divided byx+πx+πthen remainder is determined byp(−π)p(−π).

p(−π)=(−π)3+3(−π)2+3(−π)+1=(−π)3−3π2−3π+1p(−π)=(−π)3+3(−π)2+3(−π)+1=(−π)3−3π2−3π+1

Thus, the remainder is(−π)3−3π2−3π+1(−π)3−3π2−3π+1.

vv

The given polynomial isx3+3x2+3x+1x3+3x2+3x+1.

By remainder theorem, ifp(x)=x3+3x2+3x+1p(x)=x3+3x2+3x+1is divided by5+2x5+2xthen remainder is determined byp(−52)p(−52).

p(−52)=(−52)3+3(−52)2+3(−52)+1=−1258+3×254−3×52+1=−125+150−60+88=−278p(−52)=(−52)3+3(−52)2+3(−52)+1=−1258+3×254−3×52+1=−125+150−60+88=−278

Thus, the remainder is−278−278.