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Roots of Polynomials

Every polynomial with real coefficients can be written as a product of linear or quadratic terms, each having real coefficients. Proof:...

Posted by
Beetle B.
on
Thu 09 December 2021

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Descarte’s Rule of Signs

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...

Posted by
Beetle B.
on
Thu 09 December 2021

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\(z^{n}=1\)

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...

Posted by
Beetle B.
on
Wed 03 November 2021

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Roots of Quadratic Equations

Given \(x^{2}+bx+c=0\), let the roots be \(\alpha,\beta\). Then: \begin{equation*} \alpha+\beta=-b \end{equation*} \begin{equation*}...

Posted by
Beetle B.
on
Wed 03 November 2021

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Roots of Equations

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...

Posted by
Beetle B.
on
Thu 28 October 2021

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Roots of Equations In More Than One Variable

\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...

Posted by
Beetle B.
on
Thu 28 October 2021