(
a
±
b
)
2
=
a
2
±
2
a
b
+
b
2
{\displaystyle (a\pm b)^{2}=a^{2}\pm 2ab+b^{2}}
a
2
−
b
2
=
(
a
+
b
)
(
a
−
b
)
{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}
(
a
+
b
+
c
)
2
=
a
2
+
b
2
+
c
2
+
2
a
b
+
2
a
c
+
2
b
c
{\displaystyle \left(a+b+c\right)^{2}=a^{2}+b^{2}+c^{2}+2ab+2ac+2bc}
(
a
±
b
)
3
=
a
3
±
3
a
2
b
+
3
a
b
2
±
b
3
{\displaystyle (a\pm b)^{3}=a^{3}\pm 3a^{2}b+3ab^{2}\pm b^{3}}
a
3
±
b
3
=
(
a
±
b
)
(
a
2
∓
a
b
+
b
2
)
{\displaystyle a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})}
(
a
+
b
+
c
)
3
=
a
3
+
b
3
+
c
3
+
3
a
2
b
+
3
a
2
c
+
3
a
b
2
+
3
a
c
2
+
3
b
2
c
+
3
b
c
2
+
6
a
b
c
{\displaystyle \left(a+b+c\right)^{3}=a^{3}+b^{3}+c^{3}+3a^{2}b+3a^{2}c+3ab^{2}+3ac^{2}+3b^{2}c+3bc^{2}+6abc}
правіць
(
a
±
b
)
4
=
a
4
±
4
a
3
b
+
6
a
2
b
2
±
4
a
b
3
+
b
4
{\displaystyle (a\pm b)^{4}=a^{4}\pm 4a^{3}b+6a^{2}b^{2}\pm 4ab^{3}+b^{4}}
a
4
−
b
4
=
(
a
−
b
)
(
a
+
b
)
(
a
2
+
b
2
)
{\displaystyle a^{4}-b^{4}=(a-b)(a+b)(a^{2}+b^{2})}
a
2
−
b
2
{\displaystyle a^{2}-b^{2}}
a
n
−
b
n
=
(
a
−
b
)
(
a
n
−
1
+
a
n
−
2
b
+
a
n
−
3
b
2
+
…
+
a
2
b
n
−
3
+
a
b
n
−
2
+
b
n
−
1
)
{\displaystyle a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^{2}+\ldots +a^{2}b^{n-3}+ab^{n-2}+b^{n-1})}
a
2
n
−
b
2
n
=
(
a
+
b
)
(
a
2
n
−
1
−
a
2
n
−
2
b
+
a
2
n
−
3
b
2
−
…
−
a
2
b
2
n
−
3
+
a
b
2
n
−
2
−
b
2
n
−
1
)
{\displaystyle a^{2n}-b^{2n}=(a+b)(a^{2n-1}-a^{2n-2}b+a^{2n-3}b^{2}-\ldots -a^{2}b^{2n-3}+ab^{2n-2}-b^{2n-1})}
n
∈
N
{\displaystyle n\in N}
a
2
n
−
b
2
n
=
(
a
n
+
b
n
)
(
a
n
−
b
n
)
{\displaystyle a^{2n}-b^{2n}=(a^{n}+b^{n})(a^{n}-b^{n})}
a
2
n
+
1
+
b
2
n
+
1
=
(
a
+
b
)
(
a
2
n
−
a
2
n
−
1
b
+
a
2
n
−
2
b
2
−
…
+
a
2
b
2
n
−
2
−
a
b
2
n
−
1
+
b
2
n
)
{\displaystyle a^{2n+1}+b^{2n+1}=(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}b^{2}-\ldots +a^{2}b^{2n-2}-ab^{2n-1}+b^{2n})}
n
∈
N
{\displaystyle n\in N}
правіць
(
a
−
b
)
2
n
=
(
b
−
a
)
2
n
{\displaystyle (a-b)^{2n}=(b-a)^{2n}}
n
∈
N
{\displaystyle n\in N}
(
a
−
b
)
2
n
+
1
=
−
(
b
−
a
)
2
n
+
1
{\displaystyle (a-b)^{2n+1}=-(b-a)^{2n+1}}
n
∈
N
{\displaystyle n\in N}
М. Я. Выгодский. Справочник по элементарной математике. — Москва, 1958. 
