Sums and Products of Roots

In short: The roots of a polynomial are linked to its coefficients. For $a x^{2} + b x + c = 0$ , the roots add to $- \frac{b}{a}$ and multiply to $\frac{c}{a}$ . For a cubic, the sum is $- \frac{b}{a}$ , the sum of pairs is $\frac{c}{a}$ and the product is $- \frac{d}{a}$ .

Starting point: what you already know

Expanding $(x - 3) (x - 7) = x^{2} - 10 x + 21$ .
Solving a quadratic by factorising or the quadratic formula.
From the factor theorem: if the roots are $α$ and $β$ , then $(x - α)$ and $(x - β)$ are factors.
The Greek letters $α$ (alpha), $β$ (beta), $γ$ (gamma) and $δ$ (delta) are used as names for roots.

Quadratics

Suppose $a x^{2} + b x + c = 0$ has roots $α$ and $β$ . Then

$a x^{2} + b x + c = a (x - α) (x - β) = a (x^{2} - (α + β) x + α β)$

Matching the coefficients:

Check with the detective: $x^{2} - 10 x + 21$ has $a = 1$ , $b = - 10$ , $c = 21$ . Sum $= 10$ , product $= 21$ .

Cubics

For $a x^{3} + b x^{2} + c x + d = 0$ with roots $α$ , $β$ , $γ$ , expanding $a (x - α) (x - β) (x - γ)$ gives:

The middle one is the sum of the roots taken two at a time.

Quartics

For $a x^{4} + b x^{3} + c x^{2} + d x + e = 0$ with roots $α, β, γ, δ$ :

$\sum α = - \frac{b}{a}, \sum α β = \frac{c}{a}, \sum α β γ = - \frac{d}{a}, α β γ δ = \frac{e}{a}$

Building other expressions

Many expressions can be rewritten using sums and products.

$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$
For a cubic: $α^{2} + β^{2} + γ^{2} = (α + β + γ)^{2} - 2 (α β + β γ + α γ)$

Sums and products of roots

Quadratic

Sum = -b/a
Product = c/a

Cubic

Sum = -b/a
Pairs = c/a
Product = -d/a

Quartic

Signs go minus, plus, minus, plus

Uses

α² + β² = (α + β)² - 2αβ
1/α + 1/β = (α + β)/αβ
Find unknown coefficients

Year 11Using extra information about the roots

$x^{3} - 6 x^{2} + k x - 6 = 0$ has roots in the ratio $1 : 2 : 3$ . Write them as $m$ , $2 m$ , $3 m$ .

Sum: $6 m = 6$ , so $m = 1$ . The roots are 1, 2, 3.

Then $k = α β + β γ + α γ = 2 + 6 + 3 = 11$ . Check the product: $1 \times 2 \times 3 = 6 = - \frac{- 6}{1}$ . Good.

Two roots are opposites. Write the roots as $α$ , $- α$ and $β$ . The sum is just $β$ , so the sum formula gives one root at once. For $x^{3} - 3 x^{2} + k x + 12 = 0$ : $β = 3$ , and putting $x = 3$ into the equation gives $27 - 27 + 3 k + 12 = 0$ , so $k = - 4$ .

Using the polynomial itself. Each root satisfies the equation, so for $x^{3} - 2 x^{2} + x + 5 = 0$ , $α^{3} = 2 α^{2} - α - 5$ . Adding the three versions gives $\sum α^{3} = 2 \sum α^{2} - \sum α - 15$ .

Products like $(α + 1) (β + 1) (γ + 1)$ . A monic cubic is $P (x) = (x - α) (x - β) (x - γ)$ , so $P (- 1) = - (α + 1) (β + 1) (γ + 1)$ . Work out $P (- 1)$ and change the sign.

Quartic sums of squares. $\sum α^{2} = (\sum α)^{2} - 2 \sum α β$ works for any degree.

Quick summary

Quadratic: $α + β = - \frac{b}{a}$ and $α β = \frac{c}{a}$ .
Cubic: sum $- \frac{b}{a}$ , pairs $\frac{c}{a}$ , product $- \frac{d}{a}$ .
Quartic: the signs go minus, plus, minus, plus.
Rewrite other expressions, such as $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ .

Quiz

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Try 3 questions now

Two numbers add to 9 and multiply to 20. They are the roots of:
1. A $x^{2} + 9 x + 20 = 0$
2. B $x^{2} - 9 x + 20 = 0$
3. C $x^{2} - 20 x + 9 = 0$
4. D $x^{2} + 9 x - 20 = 0$
Show answer
B: $x^{2} - 9 x + 20 = 0$
$x^{2} - (sum) x + (product) = 0$ .
For $x^{2} - 5 x + 6 = 0$ , find $α + β$ .
1. A $- 5$
2. B 5
3. C 6
4. D $- 6$
Show answer
B: 5
$- \frac{b}{a} = 5$ .
For $3 x^{2} + 6 x - 2 = 0$ , find $α β$ .
1. A 2
2. B $- 2$
3. C $- \frac{2}{3}$
4. D $\frac{2}{3}$
Show answer
C: $- \frac{2}{3}$
$\frac{c}{a} = - \frac{2}{3}$ .