WOLFRAM NOTEBOOK

Vieta's Formulas for Polynomial Roots

​
x
1
-10
x
2
-4
x
3
2
x
4
6
n
1
1
2
3
(-6+x)(-2+x)(4+x)(10+x) = 480-152x-60
2
x
+6
3
x
+
4
x
a
3
=
x
1
+
x
2
+
x
3
+
x
4
= 6
a
2
=
x
1
x
2
+
x
1
x
3
+
x
1
x
4
+
x
2
x
3
+
x
2
x
4
+
x
3
x
4
= -60
a
1
=
x
1
x
2
x
3
+
x
1
x
2
x
4
+
x
1
x
3
x
4
+
x
2
x
3
x
4
= -152
a
0
=
x
1
x
2
x
3
x
4
= 480
This Demonstration shows the relationships between the coefficients of a general polynomial
P(x)
of degree
n
and its roots. These relationships are known as Vieta's formulas. They can be used for real and complex roots, but this Demonstration considers only small integer roots.
Consider the polynomial
P(x)=
n
1
(x-
x
1
)
(x-
x
2
)(x-
x
3
)(x-
x
4
)
.
Every coefficient
a
n-k
can be obtained by
∑
1≤
i
1
<..<
i
k
≤n
x
i
1
x
i
2
…
x
i
k
=
k
(-1)
a
n-k
.
When
n
1
=2
, the polynomial is rewritten as
P(x)=
2
(x-
x
1
)
(x-
x
2
)(x-
x
3
)(x-
x
4
)=(x-
x
1
)(x-
x
2
)(x-
x
3
)(x-
x
4
)(x-
x
5
)
, with
x
5
=
x
1
.
Similarly, when
n
1
=3
, the polynomial is rewritten as
P(x)=
3
(x-
x
1
)
(x-
x
2
)(x-
x
3
)(x-
x
4
)=(x-
x
1
)(x-
x
2
)(x-
x
3
)(x-
x
4
)(x-
x
5
)(x-
x
6
)
, with
x
6
=
x
1
.

References

[1] E. W. Weisstein. "Vieta's Formulas" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/VietasFormulas.html.

Permanent Citation

D. Meliga, L. Lavagnino, S. Z. Lavagnino
​
​"Vieta's Formulas for Polynomial Roots"​
​http://demonstrations.wolfram.com/VietasFormulasForPolynomialRoots/​
​Wolfram Demonstrations Project​
​Published: February 26, 2019
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