What is the simplified form of the following expression?
(x−
y
1
)
3
+(x+
y
1
)
3
- Ax\left(x^2-\frac{3}{y^2}\right)
- B-2x\left(x^2+\frac{3}{y^2}\right)
- C2x\left(x^2-\frac{1}{y^2}\right)
- D2x\left(x^2+\frac{3}{y^2}\right)
Solution & Step-by-step Explanation
Using the algebraic identity for the sum of cubes or expanding each term:
(a−b)
3
=a
3
−3a
2
b+3ab
2
−b
3
(a+b)
3
=a
3
+3a
2
b+3ab
2
+b
3
Adding these two equations gives:
(a−b)
3
+(a+b)
3
=2a
3
+6ab
2
=2a(a
2
+3b
2
)
Let a=x and b=
y
1
. Substitute these into the formula:
(x−
y
1
)
3
+(x+
y
1
)
3
=2x(x
2
+3(
y
1
)
2
)=2x(x
2
+
y
2
3
)
(a−b)
3
=a
3
−3a
2
b+3ab
2
−b
3
(a+b)
3
=a
3
+3a
2
b+3ab
2
+b
3
Adding these two equations gives:
(a−b)
3
+(a+b)
3
=2a
3
+6ab
2
=2a(a
2
+3b
2
)
Let a=x and b=
y
1
. Substitute these into the formula:
(x−
y
1
)
3
+(x+
y
1
)
3
=2x(x
2
+3(
y
1
)
2
)=2x(x
2
+
y
2
3
)