Algebra

Polynomial Integer Root Theorem

https://www.youtube.com/watch?v=fgBBJnhbMvI There are a couple of related theorems that we interchangeably call the “integer root theorem.” One tells us how to find that integer roots of a polynomial with integer coefficients. The second tells us that all rational roots of a monic polynomial with integer coefficients are integers. We prove both results in this video.

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The Quadratic Formula

https://www.youtube.com/watch?v=t9RuFHcyQRc&t=60s The ordinary way of proving the quadratic formula $$x=frac{-bpmsqrt{b^2-4ac}}{2a}$$ is through a process called “completing the square.” But there is one step at the end which requires casework that almost everyone glosses over. In this video, we show a little-known modificiation of completing the square that avoids the need for any casework.

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Telescoping & Partial Fraction Decomposition

https://www.youtube.com/watch?v=IYZQXV3SA0I Two important techniques in the ad hoc evaluation of series are telescoping and partial fraction decomposition. The latter is useful in calculus too, as an integration technique for rational functions. We show an example here of an infinite series $$frac{1}{1cdot 2 cdot 3}+frac{1}{2cdot 3cdot 4}+cdots$$ that uses both telescoping and partial fraction decomposition in

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