# Polynomial Remainder Theorem: Elementary Proof

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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 (xk – ak). 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!

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Related Videos:

Polynomial Long Division: (x - a) is a factor of (xk - ak) PROOF: https://youtu.be/Yp6VU3CkIEA
Euclidean Division of Polynomials: Theorem and Proof: https://youtu.be/ONxn17okl5c
Polynomial Remainder Theorem: Proof + Factor Theorem: https://youtu.be/q4lwSBObkXc
Polynomials - A Simple Explanation: http://youtu.be/IHIh7Y0kStE
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y
Polynomial Long Division - Examples: http://youtu.be/7XbzCQgqBPc
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Direct Substitution for Polynomials - Simple Proof: http://youtu.be/Fnb72ERTLqY
Polynomial Long Division: Multiple Variables: http://youtu.be/vrElU5SR6Aw .

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