← Topic 1 · Algebra
AMC 10/12 · Cheatsheet
Topic 1 · Algebra
Chapter 7 · AMC 12 · 1% problems
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Newton's identity
p
k
=
e
1
p
k
−
1
−
e
2
p
k
−
2
+
⋯
+
(
−
1
)
k
−
1
k
e
k
p_k = e_1 p_{k-1} - e_2 p_{k-2} + \cdots + (-1)^{k-1} k\,e_k
p
k
=
e
1
p
k
−
1
−
e
2
p
k
−
2
+
⋯
+
(
−
1
)
k
−
1
k
e
k
Power sum from Vieta
p
2
=
e
1
2
−
2
e
2
,
p
3
=
e
1
p
2
−
e
2
p
1
+
3
e
3
p_2 = e_1^2 - 2 e_2,\;\; p_3 = e_1 p_2 - e_2 p_1 + 3 e_3
p
2
=
e
1
2
−
2
e
2
,
p
3
=
e
1
p
2
−
e
2
p
1
+
3
e
3
Cubic shortcut
r
3
+
s
3
+
t
3
−
3
r
s
t
=
(
r
+
s
+
t
)
(
r
2
+
s
2
+
t
2
−
r
s
−
r
t
−
s
t
)
r^3+s^3+t^3 - 3 rst = (r+s+t)(r^2+s^2+t^2 - rs - rt - st)
r
3
+
s
3
+
t
3
−
3
r
s
t
=
(
r
+
s
+
t
)
(
r
2
+
s
2
+
t
2
−
r
s
−
r
t
−
s
t
)