## 2.2 Remainder Theorem, Factor Theorem and the Rational-Root Theorem

#### Remainder Theorem and Factor Theorem

- Previously in 2.1 Polynomial Function Introduction and Algebraic Operations we have seen that we can write division of polynomials in the form P(x)=D(x) Q(x)+R(x).

Now we are going to set D(x)=x-a. Therefore, as \operatorname{deg}(R)<\operatorname{deg}(D), R(x) must be a constant, which we denote as R.

Rewriting the equation gives

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

- Simply set x=a gives P(x)=(a-a) Q(x)+R=0+R=R. Thus, R=P(a).

- Now we generalize D(x) into any linear polynomial, i.e. D(x)=a x+b, where a,\ b are integers. Therefore, now P(x)=(a x+b) Q(x)+R and substitute x=-\frac{b}{a} (to make D(x)=0) gives