a^9+b^9+3a^6b^3+3a^3b^6
Answers
Explanation:
The factorization of a^9 + b^9 + 3a^6 b^3 + 3a^3 b^6a
9
+b
9
+3a
6
b
3
+3a
3
b
6
is (a^3+b^3)^3(a
3
+b
3
)
3
.
Explanation:
Polynomial identity : x^3+y^3+3x^2y+3xy^2= (x+y)^3x
3
+y
3
+3x
2
y+3xy
2
=(x+y)
3
(1)
The given expression : a^9 + b^9 + 3a^6 b^3 + 3a^3 b^6a
9
+b
9
+3a
6
b
3
+3a
3
b
6
This expression can be rewritten as : \begin{gathered}=a^{3\times3}+b^{3\times3}+3a^{3\times2}b^3+3a^3b^{3\times2}\\\\=(a^3)^3 + (b^3)^3 + 3(a^3)^2 (b^3) + 3(a^3) (b^3)^2\ \ [\text{Identity in exponents\ }(a^m)^n=a^{mn}]\end{gathered}
=a
3×3
+b
3×3
+3a
3×2
b
3
+3a
3
b
3×2
=(a
3
)
3
+(b
3
)
3
+3(a
3
)
2
(b
3
)+3(a
3
)(b
3
)
2
[Identity in exponents (a
m
)
n
=a
mn
]
= (a^3+b^3)^3\ \ \ \ [\text{By using (1)}]=(a
3
+b
3
)
3
[By using (1)]
Hence, the factorization of a^9 + b^9 + 3a^6 b^3 + 3a^3 b^6a
9
+b
9
+3a
6
b
3
+3a
3
b
6
is (a^3+b^3)^3(a
3
+b
3
)
3
.
# Learn more :
Factorize : a(a+b)³-3a²b(a+b)