In this video I go over further into Euclidean Division and this time look at the theorem and algorithm for univariate (i.e. single-variable) polynomials. The theorem is very similar to that for integers which I covered in my earlier video, in that both cases require proving that the associated Euclidean Division Algorithm is valid to ultimately prove that the theorem is valid. The main difference between the two algorithms is that for the integer case each step involved adding 1 to the quotient; whereas for the polynomial case we need to apply a special value, s, to ensure that the degree of the remainder keeps decreasing incrementally. The Euclidean Theorem for Division of Polynomials is that for any two univariate polynomials a and b, where b is not equal to 0 (to avoid the case of a/0 or dividing by zero) there are associated polynomials q and r called the quotient and remainder, respectively, such that a = bq + r and the degree of r is less than the degree of b. Note also that in case the remainder is 0 and the degree of b is 0, deg(0) is defined as negative. The Division Algorithm involves starting from q = 0 and then applying the theorem formula to obtain r and checking if the degree of r is less than the degree of b. If it is, then we are done. If it is not, then we add a special polynomial s, which I show is such that involves the leading coefficients of both the remainder and the divisor, b, as well as their degrees (or number of their highest power). The selection of s is such that the remainder keeps subtracting by 1 its degree until ultimately it is less than the degree of b. Thus ultimately the algorithm is proved and thus so is the theorem! The theorem and associated algorithm is the basis for polynomial long division where a/b = q + r/b and can be rearranged so that theorem holds true a = bq + r. This is a truly fascinating concept and make sure to walk-through both the algorithm as well as several examples to make Euclidean Division as clear as possible, so make sure to watch this video! Video notes and playlist: - PDF notes: https://1drv.ms/b/s!As32ynv0LoaIh4EItHHqj75MMNeJbQ - HIVE notes: https://peakd.com/mathematics/@mes/euclidean-division-of-polynomials-theorem-and-proof - Polynomial Long Division playlist: https://www.youtube.com/playlist?list=PL599994D41692CF90 Related Videos: Euclidean Division of Integers: Theorem and Proof: https://youtu.be/66juubotzi0 Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg Long Division by Hand - An in depth look: http://youtu.be/giBZg5Vqryo Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y . ------------------------------------------------------ Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I do a debunk of Eric Dubay's debunk of my earlier Flat Earth debunk and show that the Mountain's shadow forms opposite triangular shapes depending on if it is on top or below the clouds. When the Sun goes below the horizon, it can form an epic shadow of large objects such as Mountain's or skyscrapers unto the underside of the clouds. In this case, the shadow has a triangular shape where the pointy part is near the peak of the Mountain and the wider base is on the far side away on the clouds. If the Mountain peak is above the clouds, then as the Sun approaches near the horizon, it can form a shadow above the clouds that is in reverse direction as to the one below the clouds. This is pretty straight forward stuff, but that doesn't stop professional liar Eric Dubay from lying about what our eyes clearly see. I debunk Eric's many lies in methodical fashion. Video links: - MES Links: https://mes.fm/links - Debunking Flat Earth Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUyOvHJiyX74GmJ79exCkh - Debunking Flat Earth: Mountain's Shadow on Clouds: https://youtu.be/j6Ey7zUpoos - Thumbnail: https://t.me/meslinks/22541 - Shadow on Clouds: https://flatearth.ws/shadow-on-clouds - Professional liar Eric Dubay pushing bogus disinfo: https://youtu.be/xDH6a3TJqpA - Lie 1: Alludes to the Sun reflecting to ground as a possible explanation. - Lie 2: Says ALL Mountain cloud shadow images are when mountain peak is above clouds. - Lie 3: Says shadow is from above the clouds. (Clouds are in REVERSE direction) - Lie 4: Says Sun never goes below the horizon. - Why is the top of the mountain illuminated but below it is not? What is causing that shadow in FRONT of the Mountain? Is the Sun below the clouds (answer is yes). - Lie 5: He shows the clouds in REVERSE direction to what it should be, confirming he is deliberately misleading his audience aka lying. - Lie 6: His graphic confirms that he is ignoring the shadow on the front of the Mountain. - Continues lying at end of video etc. - Good debunk: Mount Rainier's Shadow and Electroflerfspectivism. https://youtu.be/-MhWmD05dN0 - Many photos of Mountain peak CLEARLY below clouds: https://presearch.com/images?q=mountain%27s+shadow+on+clouds - Mountain Shadow Above and Below Clouds Experiments: https://youtu.be/sAhQN86ZBy4 - DRAFT Experiments: https://www.youtube.com/playlist?list=PLai3U8-WIK0FfZ_7hUuyO7xN8lp5oaDiu - Mount Rainier Epic photo: https://twitter.com/KClarkWx/status/1325837129851703298 Timestamps: - Intro and Links: 0:00 - MES Links: 1:40 - Previous video: Mountain's shadow on clouds: 2:10 - Debunking Eric Dubay: 3:07 - Lie 1: Eric suggesting shadow reflections as an explanation: 3:41 - Lie 2: Eric states ALL images have Mountain peak above clouds: 4:40 - Lie 3: Eric says shadow is above the clouds: 6:06 - Lie 4: Eric says Sun never goes below the horizon: 6:52 - Top of the mountain is illuminated but not below it: 7:18 - Lie 6: Eric confirms that he is ignoring the shadow IN FRONT of the Mountain: 8:23 - Lie 5: Eric shows clouds in REVERSE direction to what it should be: 9:16 - Good debunk by Mike the Engineer: 10:22 - MES Experiment: Mountain shadow above and below clouds: 11:43 - Draft MES Experiments and Outro: 15:17 . ------------------------------------------------------ Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps