Polynomial Integer Root Theorem 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.

Polynomial Rational Root Theorem The rational root theorem gives us a finite process for finding all rational roots of a polynomial with integer coefficients. We prove the correctness of the procedure and describe how to use it.

Why is $\deg(0)=-\infty$? It is a standard convention that the degree of the zero polynomial is negative infinity. That is, $$deg(0)=-infty.$$ We explore the reasons for this strange definition, justify it, and look at some of its implications in this video.

Square Root of a Complex Number We show how to explicity compute the square roots of a complex number that is given in rectangular form. We perform the derivation in a motivated way, instead of simply mechanically verifying the formula.

The Quadratic Formula 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.

$n^{\text{th}}$ Roots of a Complex Number How can we find all of the complex $n^{text{th}}$ roots of a complex number, if the complex number is given in polar or trigonometric form? We show how to achieve this goal in our video.

De Moivre’s Formula De Moivre’s formula allows us to conveniently compute the powers of complex numbers that are written in polar or trigonometric form. We prove de Moivre in this video, which states that $$(re^{itheta})^n = r^n e^{i(ntheta)}.$$

Complex Triangle Inequality Similar to the real number triangle inequality that uses the absolute value, there is a triangle inequality for complex numbers that uses the complex modulus $$|z+w|le |z|+|w|.$$ We show how to prove this inequality here.

Telescoping & Partial Fraction Decomposition 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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