VIETA'S FORMULAS

Scroll to the bottom to see how these formulas are derived.



These formulas, which demonstrate the connection between the coefficients of a polynomial and its roots are named after the French mathematician François Vièta (1540 - 1603). These formulas may be used to check your calculations after you have solved the roots of an equation.


Quadratic Equations

Let's take this quadratic equation as an example:
2x² + 4x -6 = 0

Its 2 roots are X1 = 1 and X2 = -3
and its 3 coefficients are a = 2   b = 4   and   c = -6

For a quadratic equation, Vieta's 2 formulas state that:
X1 + X2 = -(b / a)   and   X1 * X2 = (c / a)

Now we fill the left side of the formulas with the equation's roots and the right side of the formulas with the equation's coefficients.
1 -3 = -(4 / 2)   and   1 * -3 = (-6 / 2)


Cubic Equations

Let's take this cubic equation as an example:
2x³ -4x² -22x +24 = 0

Its 3 roots are   X1 = 4   X2 = -3   X3 = 1
and its 4 coefficients are   a = 2   b = -4   c = -22   d = 24

Let's state Vieta's 3 formulas for cubic equations, and then
fill the left side of the formulas with the equation's roots and the right side of the formulas with the equation's coefficients.

X1 + X2 + X3 = -(b / a)
4 -3 + 1 = -(-4 / 2)

(X1 * X2) + (X1 * X3) + (X2 * X3) = (c/a)
(4 *-3) + (4 *1) + (-3 * 1) = (-22 / 2)

X1 * X2 * X3 = -(d/a)
4 * -3 * 1 = -(24 / 2)


Quartic Equations

Here's a quartic equation to use as an example:
3x⁴ 6x³ -123x² -126x +1,080 = 0

Its 4 roots are   X1 = 5     X2 = 3    X3 = -4   X4 = -6
and its 5 coefficients are   a = 3   b = 6   c = -123   d = -126   e = 1,080

Let's state Vieta's 4 formulas for quartic equations, and then
fill the left side of the formulas with the equation's roots and the right side of the formulas with the equation's coefficients.

X1 + X2 + X3 + X4 = -(b / a)
5 + 3 -4 -6 = -(6 / 3)

(X1 * X2) + (X1 * X3) + (X1 * X4) + (X2 * X3) + (X2 *X4) + (X3 * X4) = (c / a)
(5 * 3) + (5 * -4) + (5 * -6) + (3 * -4) + (3 * -6) + (-4 * -6) = (-123 / 3)

(X1 * X2 * X3) + (X1 * X2 * X4) + (X1 * X3 * X4) + (X2 * X3 * X4) = -(d / a)
(5 * 3 * -4) + (5 * 3 * -6) + (5 * -4 * -6) + (3 * -4 * -6) = -(-126 / 3)

X1 * X2 * X3 * X4 = (e / a)
5 * 3 * -4 * -6 = (1,080 / 3)


Quintic Equations

Here's a quintic equation to use as an example:
2x⁵ +40x⁴ +310x³ +1,160x² +2,088x +1,440 = 0

Its 5 roots are   X1 = -2     X2 = -3    X3 = -4   X4 = -5   X5 = -6
and its 6 coefficients are   a = 2   b = 40   c = 310   d = 1,160   e = 2,088   f = 1,440

Let's state Vieta's 5 formulas for quintic equations, and then
fill the left side of the formulas with the equation's roots and the right side of the formulas with the equation's coefficients.

X1 + X2 + X3 + X4 + X5 = -(b / a)
-2 -3 -4 -5 -6 = -(40 / 2)

(X1 * X2) + (X1 * X3) + (X1 * X4) + (X1 * X5) + (X2 * X3) + (X2 *X4) + (X2 *X5) + (X3 * X4) + (X3 * X5) + (X4 * X5) = (c / a)
(-2 * -3) + (-2 * -4) + (-2 * -5) + (-2 * -6) + (-3 * -4) + (-3 * -5) + (-3 * -6) + (-4 * -5) + (-4 * -6) + (-5 * -6) = (310 / 2)

(X1 * X2 * X3) + (X1 * X2 * X4) + (X1 * X2 * X5) + (X1 * X3 * X4) + (X1 * X3 * X5) + (X1 * X4 * X5) + (X2 * X3 * X4) + (X2 * X3 * X5) + (X2 * X4 * X5) + (X3 * X4 * X5) = -(d / a)
(-2 * -3 * -4) + (-2 * -3 * -5) + (-2 * -3 * -6) + (-2 * -4 * -5) + (-2 * -4 * -6) + (-2 * -5 * -6) + (-3 * -4 * -5) + (-3 * -4 * -6) + (-3 * -5 * -6) + (-4 * -5 * -6) = -(1,160 / 2)

(X1 * X2 * X3 * X4) + (X1 * X2 * X3 * X5) + (X1 * X3 * X4 * X5) + (X1 * X2 * X4 * X5) + (X2 * X3 * X4 * X5) = (e / a)
(-2 * -3 * -4 * -5) + (-2 *-3 * -4 * -6) + (-2 * -4 * -5 * -6) + (-2 * -3 * -5 * -6) + (-3 * -4 * -5 * -6) = (2,088 / 2)

X1 * X2 * X3 * X4 * X5 = -(f / a)
(-2) * (-3) * (-4) * (-5) * (-6) = -(1,440 / 2)


Sextic Equations

Here's a sextic equation to use as an example:
3x⁶ +9x⁵ -195x⁴ -405x³ +3,432x² +3,636x -15,120 = 0

Its 6 roots are   X1 = 2     X2 = -3    X3 = 4   X4 = -5   X5 = 6   X6 = -7
and its 7 coefficients are   a = 3   b = 9   c = -195   d = -405   e = 3,432   f = 3,636   g = -15,120

Let's state Vieta's 6 formulas for sextic equations, and then
fill the left side of the formulas with the equation's roots and the right side of the formulas with the equation's coefficients.

X1 + X2 + X3 + X4 + X5 + X6 = -(b / a)
2 -3 +4 -5 +6 -7 = -(9 / 3)

(X1 * X2) + (X1 * X3) + (X1 * X4) + (X1 * X5) + (X1 * X6) + (X2 * X3) + (X2 *X4) + (X2 *X5) + (X2 *X6) + (X3 * X4) + (X3 * X5) + (X3 * X6) + (X4 * X5) + (X4 * X6) + (X5 * X6) = (c / a)
(2 * -3) + (2 * 4) + (2 * -5) + (2 * 6) + (2 * -7) + (-3 * 4) + (-3 * -5) + (-3 * 6) + (-3 * -7) + (4 * -5) + (4 * 6) + (4 * -7) + (-5 * 6) + (-5 * -7) + (6 * -7) = (-195 / 3)

(X1 * X2 * X3) + (X1 * X2 * X4) + (X1 * X2 * X5) + (X1 * X2 * X6) + (X1 * X3 * X4) + (X1 * X3 * X5) + (X1 * X3 * X6) + (X1 * X4 * X5) + (X1 * X4 * X6) + (X1 * X5 * X6) + (X2 * X3 * X4) + (X2 * X3 * X5) + (X2 * X3 * X6) + (X2 * X4 * X5) + (X2 * X4 * X6) + (X2 * X5 * X6) + (X3 * X4 * X5) + (X3 * X4 * X6) + (X3 * X5 * X6) + (X4 * X5 * X6) = -(d / a)
(2 * -3 * 4) + (2 * -3 * -5) + (2 * -3 * 6) + (2 * -3 * -7) + (2 * 4 * -5) + (2 * 4 * 6) + (2 * 4 * -7) + (2 * -5 * 6) + (2 * -5 * -7) + (2 * 6 * -7) + (-3 * 4 * -5) + (-3 * 4 * 6) + (-3 * 4 * -7) + (-3 * -5 * 6) + (-3 * -5 * -7) + (-3 * 6 * -7) + (4 * -5 * 6) + (4 * -5 * -7) + (4 * 6 * -7) + (-5 * 6 * -7) = -(-405 / 3)

(X1 * X2 * X3 * X4) + (X1 * X2 * X3 * X5) + (X1 * X2 * X3 * X6) + (X1 * X2 * X4 * X5) + (X1 * X2 * X4 * X6) + (X1 * X2 * X5 * X6) + (X1 * X3 * X4 * X5) + (X1 * X3 * X4 * X6) + (X1 * X3 * X5 * X6) + (X1 * X4 * X5 * X6) + (X2 * X3 * X4 * X5) + (X2 * X3 * X4 * X6) + (X2 * X3 * X5 * X6) + (X2 * X4 * X5 * X6) + (X3 * X4 * X5 * X6) = (e / a)
(2 * -3 * 4 * -5) + (2 * -3 * 4 * 6) + (2 * -3 * 4 * -7) + (2 * -3 * -5 * 6) + (2 * -3 * -5 * -7) + (2 * -3 * 6 * -7) + (2 * 4 * -5 * 6) + (2 * 4 * -5 * -7) + (2 * 4 * 6 * -7) + (2 * -5 * 6 * -7) + (-3 * 4 * -5 * 6) + (-3 * 4 * -5 * -7) + (-3 * 4 * 6 * -7) + (-3 * -5 * 6 * -7) + (4 * -5 * 6 * -7) = (3,432 / 3)

(X1 * X2 * X3 * X4 * X5) + (X1 * X2 * X3 * X4 * X6) + (X1 * X2 * X3 * X5 * X6) + (X1 * X2 * X4 * X5 * X6) + (X1 * X3 * X4 * X5 * X6) + (X2 * X3 * X4 * X5 * X6) = -(f / a)
(2 * -3 * 4 * -5 * 6) + (2 * -3 * 4 * -5 * -7) + (2 * -3 * 4 * 6 * -7) + (2 * -3 * -5 * 6 * -7) + (2 * 4 * -5 * 6 * -7) + (-3 * 4 * -5 * 6 * -7) = -(3,636 / 3)

(X1 * X2 * X3 * X4 * X5 * X6) = (g / a)
(2 * -3 * 4 * -5 * 6 * -7) = (-15,120 / 3)


* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Should you need to determine Vieta's Formulas for other equations, the following information should be very helpful.

Quadratic Equations (Second Degree Polynomials)
Left Side of EquationRight Side
Sum of the 2 Roots=  -(b / a)
Product of the 2 Roots=  (c / a)

Cubic Equations (Third Degree Polynomials)
Sum of all 3 Roots=  -(b / a)
C (3, 2)
Sum of the 3 possible 2-term products=
  (c / a)
Product of all 3 Roots=  -(d / a)

Quartic Equations (Fourth Degree Polynomials)
Sum of all 4 Roots=  -(b / a)
C (4, 2)
Sum of the 6 possible 2-term products=
  (c / a)
C (4, 3)
Sum of the 4 possible 3-term products=
  -(d / a)
Product of all 4 Roots=  (e / a)

Quintic Equations (Fifth Degree Polynomials)
Sum of all 5 Roots=  -(b / a)
C (5, 2)
Sum of the 10 possible 2-term products=
  (c / a)
C (5, 3)
Sum of the 10 possible 3-term products=
  -(d / a)
C (5, 4)
Sum of the 5 possible 4-term products=
  (e / a)
Product of all 5 Roots=  -(f / a)

Sextic Equations (Sixth Degree Polynomials)
Sum of all 6 Roots=  -(b / a)
C (6, 2)
Sum of the 15 possible 2-term products=
  (c / a)
C (6, 3)
Sum of the 20 possible 3-term products=
  -(d / a)
C (6, 4)
Sum of the 15 possible 4-term products=
  (e / a)
C (6, 4)
Sum of the 6 possible 5-term products=
  -(f / a)
Product of all 6 Roots=  (g / a)




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