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Toolkit 9
Cube of a difference
(
a
−
b
)
3
=
a
3
−
3
a
2
b
+
3
a
b
2
−
b
3
(a-b)^3=a^3-3a^2b+3ab^2-b^3
(
a
−
b
)
3
=
a
3
−
3
a
2
b
+
3
a
b
2
−
b
3
Proof
Using Toolkit 7,
(
a
−
b
)
2
=
a
2
−
2
a
b
+
b
2
.
(a-b)^2 = a^2 - 2ab + b^2.
(
a
−
b
)
2
=
a
2
−
2
ab
+
b
2
.
Therefore,
(
a
−
b
)
3
=
(
a
−
b
)
(
a
−
b
)
2
=
(
a
−
b
)
(
a
2
−
2
a
b
+
b
2
)
=
a
3
−
2
a
2
b
+
a
b
2
−
a
2
b
+
2
a
b
2
−
b
3
=
a
3
−
3
a
2
b
+
3
a
b
2
−
b
3
.
□
\begin{aligned} (a-b)^3 &= (a-b)(a-b)^2 \\ &= (a-b)(a^2 - 2ab + b^2) \\ &= a^3 - 2a^2b + ab^2 - a^2b + 2ab^2 - b^3 \\ &= a^3 - 3a^2b + 3ab^2 - b^3. \quad\square \end{aligned}
(
a
−
b
)
3
=
(
a
−
b
)
(
a
−
b
)
2
=
(
a
−
b
)
(
a
2
−
2
ab
+
b
2
)
=
a
3
−
2
a
2
b
+
a
b
2
−
a
2
b
+
2
a
b
2
−
b
3
=
a
3
−
3
a
2
b
+
3
a
b
2
−
b
3
.
□