Question

The remainder obtained when the polynomial 1+x+x^3+x^9+x^27+x^81+x^243 is divisible by x−1 is

Answer 1

Answer:

What is the remainder when the polynomial x+x3+x9+x27+x81+x243 is divided by x2−1 ?

Dell Precision mobile workstations.

Let

P(x)=x+x3+x9+x27+x81+x243

When we divide P(x) by x2−1 we get a remainder of degree 1⟹r(x)=ax+b . This translates to

P(x)=(x+1)(x−1)q(x)+ax+b

For some polynomial q . We have that P(1)=6 and P(−1)=−6, thus

⎧⎩⎨P(1)=(1+1)(1−1)q(1)+a⋅1+bP(−1)=(−1+1)(−1−1)q(−1)+a(−1)+b

⟹{a+b=6−a+b=−6

⟹a=6,b=0 . So the remainder is 6x