Sums and Products of Roots

Starting point: what you already know

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

Quadratics

Suppose $a{x}^2+bx+c=0$ has roots $\alpha$ and $\beta$. Then

$$a{x}^2+bx+c=a(x-\alpha )(x-\beta )=a(x^2-(\alpha +\beta )x+\alpha \beta )$$

Matching the coefficients:

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

Cubics

For $a{x}^3+b{x}^2+cx+d=0$ with roots $\alpha$, $\beta$, $\gamma$, expanding $a(x-\alpha )(x-\beta )(x-\gamma )$ gives:

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

Quartics

For $a{x}^4+b{x}^3+c{x}^2+dx+e=0$ with roots $\alpha ,\beta ,\gamma ,\delta$:

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

Building other expressions

Many expressions can be rewritten using sums and products.

• ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
• $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }$
• For a cubic: ${\alpha }^2+{\beta }^2+{\gamma }^2=(\alpha +\beta +\gamma {)}^2-2(\alpha \beta +\beta \gamma +\alpha \gamma )$

Quick summary

• Quadratic: $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$.
• 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 ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$.

Quiz