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Toolkit 27
Expansion (or factorization) of two binomials
a
2
+
a
b
+
a
c
+
b
c
=
(
a
+
b
)
(
a
+
c
)
a^2+ab+ac+bc=(a+b)(a+c)
a
2
+
ab
+
a
c
+
b
c
=
(
a
+
b
)
(
a
+
c
)
Proof
Using the distributive law,
(
a
+
b
)
(
a
+
c
)
=
a
2
+
a
c
+
a
b
+
b
c
=
a
2
+
a
b
+
a
c
+
b
c
.
□
\begin{aligned} (a+b)(a+c) &= a^2+ac+ab+bc \\ &= a^2+ab+ac+bc. \quad\square \end{aligned}
(
a
+
b
)
(
a
+
c
)
=
a
2
+
a
c
+
ab
+
b
c
=
a
2
+
ab
+
a
c
+
b
c
.
□