Factorise x to the power 12 is equal to 1
Answers
Step-by-step explanation:
Break the problem down to the exponents you can handle. Ideally, Difference of Squares or sum and difference of cubes can be used. An explanation is given below on how to go about it.
Explanation:
Factoring
x
12
−
1
can be tricky but can be broken down and worked it out step by step.
Let us recollect some rules.
Difference of squares
a
2
−
b
2
=
(
a
−
b
)
(
a
+
b
)
Difference of cubes
a
3
−
b
3
=
(
a
−
b
)
(
a
2
+
a
b
+
b
2
)
Sum of cubes
a
3
+
b
3
=
(
a
+
b
)
(
a
2
−
a
b
+
b
2
)
Now let us come to our problem.
x
12
−
1
Let us make x^12 into something which we can recognize easily.
x
12
=
x
6
⋅
2
=
(
x
6
)
2
We have used
a
m
n
=
(
a
m
)
n
(
x
6
)
2
−
1
2
Now it is of form
a
2
−
b
2
(
x
6
−
1
)
(
x
6
+
1
)
Let us factorize
x
6
−
1
x
6
−
1
=
(
x
3
)
2
−
1
2
Rewriting
x
6
to get into
a
2
−
b
2
form
x
6
−
1
=
(
x
3
−
1
)
(
x
3
+
1
)
x
6
−
1
=
(
x
−
1
)
(
x
2
+
x
+
1
)
(
x
+
1
)
(
x
2
−
x
+
1
)
Now to factorize
(
x
6
+
1
)
x
6
+
1
=
(
x
2
)
3
+
1
3
Rewriting
x
6
to get it into
a
3
+
b
3
form
x
6
+
1
=
(
x
2
+
1
)
(
x
4
−
x
2
+
1
)
Factors of
(
x
6
−
1
)
(
x
6
+
1
)
=
(
x
−
1
)
(
x
2
+
x
+
1
)
(
x
+
1
)
(
x
2
−
x
+
1
)
(
x
2
+
1
)
(
x
4
−
x
2
+
1
)
Rearranging
(
x
−
1
)
(
x
+
1
)
(
x
2
+
1
)
(
x
2
−
x
+
1
)
(
x
2
+
x
+
1
)
(
x
4
−
x
2
+
1
)
Answer.
Answer link