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In this video I go over a second proof of the Polynomial Remainder Theorem which I derived in my earlier video but this time look at a what is sometimes referred to as a more “elementary” proof. In my earlier video I used the Euclidean Division theorem for Polynomials to show that the remainder of a polynomial f(x) divided by the polynomial (x – a) is simply equal to f(a), and hence is a constant. But in this video I take a look at a more “basic” approach, hence the term “elementary”, in deriving this very same theorem. From my last video I showed that (x – a) is a factor of the polynomial of the form (x^k – a^k). This becomes useful since the subtraction f(x) – f(a) is simply a linear combination of polynomials of that very same form. This means that f(x) – f(a) can be divided cleanly by (x – a) thus resulting in a quotient, q(x). Rearranging the resulting formulation we obtain f(x) = q(x)(x – a) + f(a) where f(a) is our remainder, thus proving the theorem! This is a very unique approach to deriving the polynomial remainder theorem and is always great to learn the many different ways of deriving the same theorem, so make sure to watch this video!
Playlist: Polynomial Remainder Theorem
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Polynomial Remainder Theorem: Proof #2
Recall from my earlier video on Euclidean Division for Polynomials that for two univariate polynomials a and b ≠ 0, there exists two unique polynomials q and r such that:
Also recall my earlier video on the Polynomial Remainder Theorem and the proof that simply used the Euclidean Division theorem.
https://en.wikipedia.org/wiki/Polynomial_remainder_theorem
Retrieved: 19 September 2017
Archive: https://archive.is/CGeKKPolynomial remainder theorem
In algebra, the polynomial remainder theorem or little Bézout's theorem[1] is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial f(x) by a linear polynomial x - a is equal to f(a).
In particular, x - a is a divisor of f(x) if and only if f(a) = 0.[2]
Now another proof we can do is as follows:
A slightly different proof, which may appear to some people as more elementary, starts with an observation that f(x) - f(a) is a linear combination of the terms of the form xk - ak, each of which is divisible by x - a since xk - ak = (x - a)(xk-1 + xk-2a + … + xak-2 + ak-1).
https://en.wikipedia.org/wiki/Elementary_proof
Retrieved: 6 October 2017
Archive: https://archive.is/vMLt9Elementary proof
In mathematics, an elementary proof is a mathematical proof that only uses basic techniques.
…
While the meaning has not always been defined precisely, the term is commonly used in mathematical jargon. An elementary proof is not necessarily simple, in the sense of being easy to understand: some elementary proofs can be quite complicated.[1]
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Nicely explained video. . . Good job !