Chapter 4.3, Problem 12CFU

(a)

To determine

To state: The degree of $f\left(x\right)$ .

Answer to Problem 12CFU

  $\text{Degree}=12$

Explanation of Solution

Given information: Let $f\left(x\right)=x^7+x^9+x^{12}-2x^2$

Calculation:

The highest power of x in the polynomial is $x^{12}$ .

Thus, degree of $f\left(x\right)$ is 12.

(b)

To determine

To state: The number of complex zeros that $f\left(x\right)$ has.

Answer to Problem 12CFU

  $\text{Number of complex zeros}=12$

Explanation of Solution

Given information: Let $f\left(x\right)=x^7+x^9+x^{12}-2x^2$

Calculation:

The number of complex roots (zeros) of a polynomial is equal to the degree of the polynomial.

Thus,

  $\text{Number of complex zeros}=12$

(c)

To determine

To state: The degree of the depressed polynomial that would result from dividing $f\left(x\right)$ by $x-a$ .

Answer to Problem 12CFU

  $\text{Degree of the depressed polynomial}=11$

Explanation of Solution

Given information: Let $f\left(x\right)=x^7+x^9+x^{12}-2x^2$

Calculation:

The degree of the depressed polynomial is one less than the degree of the polynomial.

Thus,

  $\text{Degree of the depressed polynomial}=11$

(d)

To determine

To find: One factor of $f\left(x\right)$ .

Answer to Problem 12CFU

  $\begin{array}{l}\text{Possible factors:}\\ x,x^2,x^{11}+x^8+x^6-2x\text{ or }{x}^{10}+x^7+x^5-2\end{array}$

Explanation of Solution

Given information: Let $f\left(x\right)=x^7+x^9+x^{12}-2x^2$

Calculation:

A quick look at the polynomial tells us that there is no constant term.

This means $f\left(0\right)=0$

  $\left(\text{inserting }x=0\text{ results in 0}\right).$

This means (by the Factor Theorem)

That,

  $\left(x-0\right)\text{ is a factor of }f\left(x\right).$

There, we found one, $g\left(x\right)=x$

Some more could be found by:

  $\begin{array}{l}f\left(x\right)=x^7+x^9+x^{12}-2x^2\\ f\left(x\right)=x^{12}+x^9+x^7-2x^2\\ f\left(x\right)=x\left(x^{11}+x^8+x^6-2x\right)\\ f\left(x\right)=x^2\left(x^{10}+x^7+x^5-2\right)\end{array}$

  $\begin{array}{l}\text{Possible factors:}\\ x,x^2,x^{11}+x^8+x^6-2x\text{ or }{x}^{10}+x^7+x^5-2\end{array}$