Roots of Polynomials

$$a^2+b^2\equiv (a+bi)(a-bi)$$

For a polynomial with real coefficients, complex roots come in conjugate pairs, that is: $f(z)=0$ if and only if $f(z^{\ast })=0$.

Roots and Coefficients

Let $\alpha$, $\beta$, $\gamma$, and $\delta$ be the roots of the following polynomials (as necessary according to the degree of each). The following relationships are known as Vieta’s formulae.

For a quadratic $a{x}^2+bx+c=0$:

$$\begin{array}{rl}\alpha +\beta & =\frac{-b}{a}\\ \alpha \beta & =\frac{c}{a}\end{array}$$

For a cubic $a{x}^3+b{x}^2+cx+d=0$:

$$\begin{array}{rl}\Sigma {\alpha }^{\prime }=\alpha +\beta +\gamma & =\frac{-b}{a}\\ \Sigma \alpha {\beta }^{\prime }=\alpha \beta +\alpha \gamma +\beta \gamma & =\frac{c}{a}\\ \Sigma \alpha \beta {\gamma }^{\prime }=\alpha \beta \gamma & =\frac{-d}{a}\end{array}$$

For a quartic $a{x}^4+b{x}^3+c{x}^2+dx+e=0$:

$$\begin{array}{rl}\Sigma {\alpha }^{\prime }=\alpha +\beta +\gamma +\delta & =\frac{-b}{a}\\ \Sigma \alpha {\beta }^{\prime }=\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta & =\frac{c}{a}\\ \Sigma \alpha \beta {\gamma }^{\prime }=\alpha \beta \gamma +\alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta & =\frac{-d}{a}\\ \Sigma \alpha \beta \gamma {\delta }^{\prime }=\alpha \beta \gamma \delta & =\frac{e}{a}\end{array}$$

Using Substitutions

If an equation in $x$ has a root $x=p$, a substitution $u=f(x)$ can be applied, giving a resulting equation in $u$ having a root $u=f(p)$.

Example The equation $x^2+7x+12=0$ has roots $p$ and $q$. Find a quadratic equation with roots $p^2$ and $q^2$.

$$u=x^2⟹x=\sqrt{u}$$ $$\begin{array}{rl}(\sqrt{u}{)}^2+7(\sqrt{u})+12& =0\\ u+12& =-7\sqrt{u}\\ (u+12{)}^2& =(-7\sqrt{u}{)}^2\\ u^2+24u+144& =49u\\ u^2-25u+144& =0\end{array}$$