Question

if x power 51+51 is divided by x+1 find the remainder

Answer 1

what do you mean????????????

Answer 2

The remainder is 50.

One way to see it is through smartly expanding x51+51
x
51
+
51
. The approach is as follows:

x51+51=x51+(x50−x50)+(−x49+x49)+(x48−x48)+...+(x2−x2)+(−x+x)+1+50
x
51
+
51
=
x
51
+
(
x
50
−
x
50
)
+
(
−
x
49
+
x
49
)
+
(
x
48
−
x
48
)
+
.
.
.
+
(
x
2
−
x
2
)
+
(
−
x
+
x
)
+
1
+
50

Now, regrouping the terms gives,
x51+51=(x51+x50)−(x50+x49)+(x49+x48)−x48+...+x2−(x2+x)+(x+1)+50
x
51
+
51
=
(
x
51
+
x
50
)
−
(
x
50
+
x
49
)
+
(
x
49
+
x
48
)
−
x
48
+
.
.
.
+
x
2
−
(
x
2
+
x
)
+
(
x
+
1
)
+
50

=x50(x+1)−x49(x+1)+x48(x+1)−...−x(x+1)+(x+1)+50
=
x
50
(
x
+
1
)
−
x
49
(
x
+
1
)
+
x
48
(
x
+
1
)
−
.
.
.
−
x
(
x
+
1
)
+
(
x
+
1
)
+
50

=(x+1)(x50−x49+x48...−x+1)+50
=
(
x
+
1
)
(
x
50
−
x
49
+
x
48
.
.
.
−
x
+
1
)
+
50

So, x51+51=(x+1)p(x)+50
x
51
+
51
=
(
x
+
1
)
p
(
x
)
+
50
, where p(x) = (x50−x49+x48...−x+1)
(
x
50
−
x
49
+
x
48
.
.
.
−
x
+
1
)

Now, does the above expression ring a bell about what we learnt in school which is something like:

Dividend=(Divisor∗Quotient)+Remainder
D
i
v
i
d
e
n
d
=
(
D
i
v
i
s
o
r
∗
Q
u
o
t
i
e
n
t
)
+
R
e
m
a
i
n
d
e
r

Mapping this representation to the above case results in as follows:
Dividend=x51+51
D
i
v
i
d
e
n
d
=
x
51
+
51

Divisor=x+1
D
i
v
i
s
o
r
=
x
+
1

Quotient=p(x)=(x50−x49+x48...−x+1)
Q
u
o
t
i
e
n
t
=
p
(
x
)
=
(
x
50
−
x
49
+
x
48
.
.
.
−
x
+
1
)

Remainder=50
R
e
m
a
i
n
d
e
r
=
50

Another way to see this is through the remainder theorem,
Polynomial remainder theorem

Let's say f(x)=x51+51
f
(
x
)
=
x
51
+
51
, then the remainder r
r
, obtained by dividing f(x)
f
(
x
)
with x+1=x−(−1)
x
+
1
=
x
−
(
−
1
)
is f(−1)
f
(
−
1
)
.

Hence, r=f(−1)=(−1)51+51=−1+51=50