But there's only four powers!
Find the sum of the fifth powers of the roots of the equation x 4 − 7 x 2 + 4 x − 3 = 0 .
The answer is -140.
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2 solutions
exactly what i did. upvoted.
Exactly not what I did. Up vote.
Applied Newton's identity to the equation directly.
e
1
=
0
e
2
=
−
7
e
3
=
−
4
e
4
=
−
3
P
1
=
e
1
=
0
.
P
2
=
e
1
∗
P
1
−
2
∗
e
2
=
0
−
2
∗
(
−
7
)
=
1
4
.
P
3
=
e
1
∗
P
2
−
e
2
∗
P
1
+
3
e
3
=
0
−
0
+
3
∗
(
−
4
)
=
−
1
2
P
4
i
s
m
u
l
t
i
p
l
i
e
d
.
b
y
0
P
5
=
e
1
∗
P
4
−
e
2
∗
P
3
+
e
3
∗
P
2
−
e
4
∗
P
1
=
0
−
(
−
7
)
∗
(
−
1
2
)
+
(
−
4
)
∗
1
4
+
0
=
−
1
4
0
.
Using Newton's Sum directly is a basic approach but it is a long procedure to find the sum of roots till 5 t h power. Although I'll also calculate the sum of roots till 3 r d power.
x 4 − 7 x 2 + 4 x − 3 = 0 x 5 = 7 x 3 − 4 x 2 + 3 x ∑ x 5 = ∑ ( 7 x 3 − 4 x 2 + 3 x ) = 7 ∑ x 3 − 4 ∑ x 2 + 3 ∑ x By Newton’s Sums : ∑ x = 0 , ∑ x 2 = 1 4 and ∑ x 3 = − 1 2 ∴ ∑ x 5 = 7 ( − 1 2 ) − 4 ( 1 4 ) + 3 ( 0 ) = − 1 4 0