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Remainder theorem:

If the polynomial $P(x)$ is divided by $(x-a)$, then the remainder is $P(a)$.

Factor theorem: If $P(a)=0$, then $(x-a)$ is a factor of $P(x)$

I am wondering if they are equivalent - if both imply the other, or one is a consequence of the other.

My intuition is that they are equivalent, but I am not sure how to prove. It seems clear that the remainder theorem implies the factor theorem (by definition of factor). I am not sure about the other direction.

Am I on the right track? Thank you.

Bill Dubuque
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Andrew
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    As far as I can tell, your intuition is correct in that the remainder theorem implies the factor theorem, but not the other direction. I trust somebody will explain this in more detail in an answer. – John Omielan Aug 08 '19 at 01:54
  • Factor and remainder theorem are connected such that both implies each other – user75659 Aug 08 '19 at 03:31
  • It seems to me that in either direction you have to apply some definitions and facts regarding polynomials, but then each theorem is a corollary of the other. – David K Aug 08 '19 at 11:01

2 Answers2

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Yes, both are equivalent to the Division algorithm with linear divisor $\,x\!-\!a$. Below, if rings are unfamiliar, let $R[x] = $ polynomials whose coefficients are complex (or real or rational). As for integers, $\, f\bmod g :=\,$ remainder left in $f\div g = \,$ polynomial (long) division with remainder.

Theorem $\ $ TFAE for a polynomial $\,f\in R[x],\,$ and $\,a\in R\,$ a commutative ring.

$(0)\ \ \ f = (x\!-\!a)q + r\ $ for some $\,q\in R[x],\ r\in R\ \ \ $ [Monic Linear Division Algorithm]

$(1)\ \ \ f\bmod x\!-\!a = f(a)\ \ \ \ \ \ \ $ [Remainder Theorem]

$(2)\ \ \ f(a) = 0\,\Rightarrow\, x\!-\!a\mid f\ \ \ $ [Factor Theorem]

Proof $\ (0\Rightarrow 1)\ \ \ f = (x\!-\!a)q + r\,\overset{\large x\,=\,a}\Longrightarrow\, r=f(a)\,\Rightarrow\,f\bmod x\!-\!a = r = f(a) $

$(1\Rightarrow 2)\ \ \ f\bmod x\!-\!a = f(a) = 0\,\Rightarrow\, x\!-\!a\mid f$

$(2\Rightarrow 0)\ \ \ g := f-f(a)\,$ has $\,g(a) = 0\ $ so $\ g = f-f(a) = (x\!-\!a)q$

Remark $ $(Euclidean) polynomial division with (smaller) remainder is always achievable for divisors that are monic (lead coef $\color{#c00}{= 1}$ or a $\rm\color{#c00}{unit}$ = invertible), such as $\,\color{#c00}1x-a\,$ above, see here.

Bill Dubuque
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The Remainder Theorem implies the factor theorem as noted by John; We can rewrite the statement 'If the polynomial $P(x)$ is divided by $(x-a)$, then the remainder is $P(a)$' as

$$P(x) = (x-a)Q(x)+P(a)$$

where $Q(x)$ is a polynomial of one degree less than $P(x)$. Now, if $P(a)=0$, what do you get?

BaroqueFreak
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    You can get the reverse implication also by considering factor theorem on the polynomial $P(x)-P(a)$ – Macavity Aug 08 '19 at 02:23