Sums and products of zeroes

Formulae for the sum and product of zeroes of quadratics, cubics and quartics, the general pattern, and forming identities such as the sum of squares of the zeroes.

The formulae

Expand $a(x - \alpha)(x - \beta)(x - \gamma)$ and compare its coefficients with $ax^3 + bx^2 + cx + d$. The coefficients turn out to be built from the zeroes.

Quadratic $ax^2 + bx + c$

$$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}$$

Cubic $ax^3 + bx^2 + cx + d$

$$\sum\alpha = -\frac{b}{a}, \quad \sum\alpha\beta = \frac{c}{a}, \quad \alpha\beta\gamma = -\frac{d}{a}$$

Here $\sum\alpha\beta$ means the sum of the products in pairs, $\alpha\beta + \beta\gamma + \gamma\alpha$.

Quartic ax⁴ + bx³ + cx² + dx + e

$$\sum\alpha = -\frac{b}{a}, \quad \sum\alpha\beta = \frac{c}{a}, \quad \sum\alpha\beta\gamma = -\frac{d}{a}, \quad \alpha\beta\gamma\delta = \frac{e}{a}$$

Forming identities

Example 1●●

The roots of $x^3 - 2x^2 + 3x + 5 = 0$ are $\alpha, \beta, \gamma$. Find $\alpha^2 + \beta^2 + \gamma^2$ and $\dfrac1\alpha + \dfrac1\beta + \dfrac1\gamma$.

$\sum\alpha = 2$, $\;\sum\alpha\beta = 3$, $\;\alpha\beta\gamma = -5$.

$\alpha^2 + \beta^2 + \gamma^2 = 2^2 - 2 \times 3 = -2$

(α + β + γ)² = α² + β² + γ² + 2(αβ + βγ + γα).

$\dfrac1\alpha + \dfrac1\beta + \dfrac1\gamma = \dfrac{\alpha\beta + \beta\gamma + \gamma\alpha}{\alpha\beta\gamma} = -\dfrac35$

Use a common denominator αβγ.

Using extra information to factor

Example 2●●●

The roots of $x^3 - 6x^2 + 11x - 6 = 0$ form an arithmetic sequence. Solve the equation.

Sum: $3a = 6$, so $a = 2$ is a root.

Write the roots as a − d, a, a + d.

Divide by $x - 2$: $(x - 2)(x^2 - 4x + 3) = (x - 2)(x - 1)(x - 3)$.

Roots $1, 2, 3$.

Example 3●●●

One root of $x^3 + px^2 - 4x - 12 = 0$ is the negative of another. Find $p$ and solve the equation.

Write the roots as $\alpha, -\alpha, \beta$. Pairs: $-\alpha^2 + \alpha\beta - \alpha\beta = -\alpha^2 = -4$, so $\alpha = \pm 2$.

$-\alpha^2\beta = 12$, so $-4\beta = 12$ and $\beta = -3$.

Product of the roots is −d/a = 12.

$\alpha - \alpha + \beta = -p$, so $p = 3$. The roots are $2, -2, -3$.

Sum of the roots is −p.

Flashcards

Cubic: $lpha + eta + gamma$

$-\dfrac{b}{a}$

Cubic: $lphaeta + etagamma + gammalpha$

$\dfrac{c}{a}$

Cubic: $lphaetagamma$

$-\dfrac{d}{a}$

$lpha^2 + eta^2 + gamma^2$

$\left(\sum\alpha\right)^2 - 2\sum\alpha\beta$

Practice

The roots of $x^2 - 5x + 3 = 0$ are $\alpha$ and $\beta$. Find $\alpha + \beta$.
The roots of $2x^3 - 4x^2 + x - 7 = 0$ are $\alpha, \beta, \gamma$. Find $\alpha + \beta + \gamma$.
For $2x^3 - 4x^2 + x - 7 = 0$, find $\alpha\beta\gamma$.
For $x^3 - 2x^2 + 3x + 5 = 0$, find $\alpha^2 + \beta^2 + \gamma^2$.
For $x^3 - 2x^2 + 3x + 5 = 0$, find $\dfrac1\alpha + \dfrac1\beta + \dfrac1\gamma$.
For the quartic $ax^4 + bx^3 + cx^2 + dx + e$, the product of the zeroes is:
The roots of $x^3 - 6x^2 + 11x - 6 = 0$ form an arithmetic sequence. Find the middle root.
$\alpha^2 + \beta^2 + \gamma^2$ equals:
The roots of $x^3 + kx^2 + 2x - 8 = 0$ add to $6$. Find $k$.
For a monic cubic with roots $1, 2, 3$, the constant term is: