Formula

Sum and product of the zeros of a cubic

Algebra • Polynomial • Zeros and Coefficients

Formula

α + β + γ = - \frac{b}{a}, α β + β γ + γ α = \frac{c}{a}, α β γ = - \frac{d}{a}

α

β

and

γ

are the zeros of

a x^{3} + b x^{2} + c x + d

with

a \neq = 0

, then

α + β + γ = - \frac{b}{a}

α β + β γ + γ α = \frac{c}{a}

and

α β γ = - \frac{d}{a}

How to use it

1The signs alternate as the coefficients are read downwards: $- \frac{b}{a}$ , then $+ \frac{c}{a}$ , then $- \frac{d}{a}$ .
2Any symmetric expression in the three zeros can be rebuilt from these three, so the zeros themselves never have to be found.
3To build a cubic from given zeros, write $x^{3} - (sum) x^{2} + (sum of pairs) x - (product)$ .

Worked example

The zeros of

2 x^{3} - 12 x^{2} + 22 x - 12

are

α

β

and

γ

. Find

α + β + γ

and

α β γ

Step-by-step

1
$α + β + γ = - \frac{- 12}{2} = 6$
Why: Apply $- \frac{b}{a}$ with $b = - 12$ and $a = 2$ .
2
$α β γ = - \frac{- 12}{2} = 6$
Why: Apply $- \frac{d}{a}$ with $d = - 12$ .

Answer

α + β + γ = 6

and

α β γ = 6

Watch out

Do not drop the leading coefficient: for

2 x^{3} - 12 x^{2} + \dots

the sum of the zeros is

6

, not

12

. A repeated zero is counted as many times as it repeats.

Builds on

Sum and product of roots (Vieta)

Also known as

sum of roots of a cubic, Vieta for a cubic, relation between zeros and coefficients, alpha beta gamma, symmetric functions of the roots, zeros of a cubic polynomial

Sum and product of the zeros of a cubic

Formula

Worked example

Appears in past papers

2019 (II)

2018 (II)