Question

What is the difference between the remainder theorem and the factor theorem?

Answer 1

Difference between the remainder theorem and the factor theorem.

Remainder theorem

• The Remainder Theorem states that dividing the polynomial $p\left(x\right)$ by some linear factor $x–a,$ through long polynomial division, results in some polynomial $q\left(x\right)$ . It can be expressed as:

$\frac{p\left(x\right)}{x-a}=q\left(x\right)+r\left(x\right)$

Factor theorem

• Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial. It is a special case of a polynomial remainder theorem.
• According to factor theorem, if $f\left(x\right)$ is a polynomial of degree $n\ge 1$ then, $(x-a)$ is a factor of $f\left(x\right)$, if $f\left(a\right)=0$. If $f\left(x\right)$ is a polynomial, then then, $(x-a)$ is a factor of $f\left(x\right)$, iff $f\left(a\right)=0$ ,where $a$ is the root.