MathGoldMedalist
Home
Toolkit
Problems
About
Sign In
Register
Home
Toolkit
Problems
About
←
Back to Toolkit
Toolkit 11
Difference of cubes
a
3
−
b
3
=
(
a
−
b
)
(
a
2
+
a
b
+
b
2
)
a^3-b^3=(a-b)(a^2+ab+b^2)
a
3
−
b
3
=
(
a
−
b
)
(
a
2
+
ab
+
b
2
)
Proof
Using the distributive law,
(
a
−
b
)
(
a
2
+
a
b
+
b
2
)
=
a
3
+
a
2
b
+
a
b
2
−
a
2
b
−
a
b
2
−
b
3
=
a
3
−
b
3
.
□
\begin{aligned} (a-b)(a^2 + ab + b^2) &= a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 \\ &= a^3 - b^3. \quad\square \end{aligned}
(
a
−
b
)
(
a
2
+
ab
+
b
2
)
=
a
3
+
a
2
b
+
a
b
2
−
a
2
b
−
a
b
2
−
b
3
=
a
3
−
b
3
.
□