These formulas, which demonstrate the connection between the coefficients of a polynomial and its roots are named after the French mathematician François Viète (1540 – 1603), usually referred to as “Vieta”. 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 = 24Let’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 = 1,080Let’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 = 1,440Let’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 = -15,120Let’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)

Quadratic Equations (Second Degree Polynomials)

Left Side of Equation Right 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)