Polynomial Remainder Theorem: Proof + Factor Theorem

Math Easy Solutions

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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!

Video notes and playlist:

PDF notes: https://1drv.ms/b/s!As32ynv0LoaIh4ElG5XEOZkYZP2tzw
HIVE notes: https://peakd.com/mathematics/@mes/polynomial-remainder-theorem-proof-factor-theorem
Polynomial Remainder Theorem playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EpiKxzYHig3rGsNj03354I