What are the factors of x^3- 1?
Answers
Step-by-step explanation:
Expanding upon prior answer:
Explanation:
I want to expand upon an idea expressed in the prior answer
The idea of:
x
n
−
1
x
−
1
=
n
∑
r
=
1
x
n
−
r
or not in sigma notation:
x
n
−
1
x
−
1
=
x
n
−
1
+
x
n
−
2
+
...
+
x
+
1
We can prove this via induction:
Basis case :
⇒
n
=
1
L
H
S
:
x
1
−
1
x
−
1
=
1
R
H
S
:
x
1
−
1
=
x
0
=
1
Hence basis case holds
Induction:
Assume
n
=
k
holds:
x
k
−
1
x
−
1
=
k
∑
r
=
1
x
k
−
r
n
=
k
+
1
:
k
+
1
∑
r
=
1
x
k
+
1
−
r
=
(
k
∑
r
=
1
x
k
+
1
−
r
)
+
1
=
x
⋅
(
k
∑
r
=
1
x
k
−
r
)
+
1
=
x
⋅
(
x
k
−
1
x
−
1
)
+
1
=
x
k
+
1
−
x
x
−
1
+
1
=
x
k
+
1
−
x
x
−
1
+
x
−
1
x
−
1
=
x
k
+
1
−
1
x
−
1
Hence this is also what we yield when plugging directly into formula:
Hence holds for all
k
∈
Z
+
and all
k
+
1
∈
Z
+
so holds for all
n
∈
Z
+
⇒
Proven by mathematical induction
I thought this was a nice idea to consider!
Dear friend the answer for this question is here:
x3 + 1 = x3 + 1
the sum of cubes tells us that:
a3 + b3 = (a+b) (a2 - ab + b2)
hence x3 + 1 = x+1( x2 - x.1) (+1)
=(x3 + 1)(x2-x) +1
Hope it helps you!!!