Chapter 2.3, Problem 5E

To determine

To fill : the blank space with appropriate term

Answer to Problem 5E

The $\underset{\_}{\boxed{FACTOR}}$ Theorem states that a polynomial $f\left(x\right)$ has a factor $\left(x-k\right)$ if and only if $f\left(k\right)=0$ .

Explanation of Solution

Given information : The $\_\_\_\_\_\_\_\_\_\_$ Theorem states that a polynomial $f\left(x\right)$ has a factor $\left(x-k\right)$ if and only if $f\left(k\right)=0$ .

Concept Involved:

For long division of polynomials by divisors of the form x - k , there is a shortcut

called synthetic division. The pattern for synthetic division of a cubic polynomial is

summarized below. (The pattern for higher-degree polynomials is similar.)

Synthetic Division (for a Cubic Polynomial): To divide $a{x}^3+b{x}^2+cx+d$ by $x-k$ , use this pattern.

$\begin{array}{l}\begin{array}{l}k\\ \end{array}|\stackrel{Coefficientofdividend}{\overbrace{\underset{\_}{\begin{array}{cccc}\boxed{a}& b& c& d\\ \downarrow & \underset{\downarrow }{\boxed{ka}}& \underset{\downarrow }{\boxed{}}& \underset{\downarrow }{\boxed{}}\end{array}}}}\\ \begin{array}{ccccc}& \nearrow & \nearrow & \nearrow & \end{array}\\ \underset{Coefficientofquotient}{\underbrace{\begin{array}{ccc}\boxed{a}& \boxed{}& \boxed{}\end{array}}}\begin{array}{cc}\boxed{r}& \leftarrow \text{Remainder}\end{array}\end{array}$

In case when we have a polynomial with a missing term, insert placeholders with zero coefficients for missing powers of the variable. Vertical pattern: Add terms in columns Diagonal pattern: Multiply results by k. This algorithm for synthetic division works only for divisors of the form x - k. Remember that $x+k=x-\left(-k\right)$.

The Division Algorithm : If $f\left(x\right)$ and $d\left(x\right)$ are polynomials such that $d\left(x\right)\ne 0$ , and the degree of $d\left(x\right)$ isless than or equal to the degree of $f\left(x\right)$ , then there exist unique polynomials $q\left(x\right)$ and $r\left(x\right)$ such that $f\left(x\right)=d\left(x\right)\cdot q\left(x\right)+r\left(x\right)$ where $r\left(x\right)=0$ or the degree of $r\left(x\right)$ is less than the degree of $d\left(x\right)$ . If theremainder $r\left(x\right)$ is zero, then $d\left(x\right)$ divides evenly into $f\left(x\right)$ .

Another way to write the Division Algorithm is $\frac{f\left(x\right)}{d\left(x\right)}=q\left(x\right)+\frac{r\left(x\right)}{d\left(x\right)}$ .

In the Division Algorithm, the rational expression $f\left(x\right)/d\left(x\right)$ is improper because the degree of $f\left(x\right)$ is greater than or equal to the degree of $d\left(x\right)$ . On the other hand, the rational expression $r\left(x\right)/d\left(x\right)$ is proper because the degree of $r\left(x\right)$ is less than the degree of $d\left(x\right)$ .

The Factor Theorem: A polynomial $f\left(x\right)$ has a factor $\left(x-k\right)$ if and only if $f\left(k\right)=0$ . Using the Factor Theorem, you can test whether a polynomial has $\left(x-k\right)$ as a factor by evaluating the polynomial at $x=k$ . If the result is 0, then $\left(x-k\right)$ is a factor.