the value of x6 and y6
Answers
Step-by-step explanation:
The required factored form of the given expression is (x+y)(x-y)(x^2-xy+y^2)(x^2+xy+y^2).(x+y)(x−y)(x
2
−xy+y
2
)(x
2
+xy+y
2
).
Step-by-step explanation: We are given to factorize the following expression :
E=x^6-y^6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)E=x
6
−y
6
(i)
We will be using the following factorization formulas :
\begin{gathered}(i)~a^2-b^2=(a+b)(a-b),\\\\(ii)~a^3-b^3=(a-b)(a^2+ab+b^2),\\\\(iii)~a^3+b^3=(a+b)(a^2-ab+b^2).\end{gathered}
(i) a
2
−b
2
=(a+b)(a−b),
(ii) a
3
−b
3
=(a−b)(a
2
+ab+b
2
),
(iii) a
3
+b
3
=(a+b)(a
2
−ab+b
2
).
The factorization of given expression (i) is as follows :
\begin{gathered}E\\\\=x^6-y^6\\\\=(x^3)^2-(y^3)^2\\\\=(x^3+y^3)(x^3-y^3)\\\\=(x+y)(x^2-xy+y^2)(x-y)(x^2+xy+y^2)\\\\=(x+y)(x-y)(x^2-xy+y^2)(x^2+xy+y^2).\end{gathered}
E
=x
6
−y
6
=(x
3
)
2
−(y
3
)
2
=(x
3
+y
3
)(x
3
−y
3
)
=(x+y)(x
2
−xy+y
2
)(x−y)(x
2
+xy+y
2
)
=(x+y)(x−y)(x
2
−xy+y
2
)(x
2
+xy+y
2
).
Thus, the required factored form of the given expression is (x+y)(x-y)(x^2-xy+y^2)(x^2+xy+y^2).(x+y)(x−y)(x
2
−xy+y
2
)(x
2
+xy+y
2
).