Every polynomial with real coefficients can be written as a product of linear or quadratic terms, each having real coefficients. Proof:...
If a polynomial \(f(x)\) has real coefficients, and has a nonzero constant term, then the number of positive real roots is the number of...
Say we want the roots of \(z^{n}=1\). Let \(z=re^{i\theta}\). Then \(z^{n}=r^{n}e^{in\theta}=1\). We immediately see that \(r=1\). Thus...
Given \(x^{2}+bx+c=0\), let the roots be \(\alpha,\beta\). Then: \begin{equation*} \alpha+\beta=-b \end{equation*} \begin{equation*}...
Remainder Theorem If a polynomial \(p(x)\) is divided by \(x-r\), then the remainder is \(p(r)\). To prove this, let \(p(x)=Q(x)(x-r) +...
Equations of the Form \(\sqrt{ax^{2}+bx+c}\pm\sqrt{ax^{2}+bx+d}=k\) \begin{equation*} \sqrt{ax^{2}+bx+c}+\sqrt{ax^{2}+bx+d}=k...
\begin{equation*} 2*x^{2}-xy+y^{2}=4 \end{equation*} \begin{equation*} 4*x^{2}-5xy+3y^{2}=6 \end{equation*} Manipulate the two equations...