# Polynomial Remainder Theorem: Proof + Factor Theorem

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• Published on Dec 27, 2020

In this video I go over a special case of Euclidean Division known as the Polynomial Remainder Theorem. This theorem states that the if a polynomial f(x) is divided by the linear polynomial x – a, where a is a constant, then the remainder is equal to f(a). The derivation of this theorem is actually quite simple when invoking the Euclidean Division Theorem for Polynomials, which I covered in my last video. Recall that the Euclidean Division states that for the division of two polynomials f(x)/b(x), there are two polynomials q and r such that: f(x) = b(x)q(x) + r(x) where the degree of r is less than the degree of b (or if r(x) = 0). Thus we can simply let b(x) = (x – a) which clearly shows that f(a) = (a – a) * q + r = r(x). Thus we have proved the theorem.

Also in this video I go over some other useful insights into this theorem such as the case when the remainder, r, is equal to 0, which thus makes (x – a) a divisor or factor of f(x). This makes it useful in simplifying polynomial division and is the basis of the Factor Theorem. The Factor Theorem states that the linear polynomial (x – a) is a factor of the polynomial f(x) if and only if f(a) = 0. I may elaborate further into this theorem so stay tuned (and let me know if I should)!

I also go over a simple example to illustrate the Polynomial Remainder Theorem. This is a very interesting application of the Euclidean Division Theorem and is useful in computational methods of long division so make sure to watch this video!

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/polynomial-remainder-theorem-proof-factor-theorem

Related Videos:

Euclidean Division of Polynomials: Theorem and Proof: https://youtu.be/ONxn17okl5c
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 .

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