Chapter 3, Problem 12RCC

a.

To determine

To state the fundamental theorem of algebra.

Explanation of Solution

Given information:

The given statement is:

“State the Fundamental Theorem of Algebra”

The Fundamental Theorem of Algebra states that every polynomial with complex coefficients has at least one complex zero.

  $\text{P}\left(\text{x}\right){\text{=a}}_{\text{n}}{\text{x}}^{\text{n}}{\text{+a}}_{\text{n}-\text{1}}{\text{x}}^{\text{n}-\text{1}}\text{+}\mathrm{...}{\text{+a}}_{\text{1}}{\text{x+a}}_{\text{0}}\left(\text{n}\ge 1,{\text{a}}_{\text{n}}\ne 0\right)$

b.

To determine

To state the complete factorization theorem.

Explanation of Solution

Given information:

The given statement is:

“State the Complete Factorization Theorem”

If $\text{P}\left(\text{x}\right)$ is a polynomial of degree $\text{n}\ge 1$ , there exists complex numbers ${\text{a,c}}_{\text{1}}{\text{,c}}_{\text{2}}\mathrm{...}{\text{c}}_{\text{n}}\left(\text{a}\ne \text{0}\right)$ such that

  $\text{P}\left(\text{x}\right)\text{=a}\left(\text{x}-{\text{c}}_{\text{1}}\right)\left(\text{x}-{\text{c}}_{\text{2}}\right)\mathrm{...}\left(\text{x}-{\text{c}}_{\text{n}}\right)$

c.

To determine

To determine the meaning of the statement “ $\text{c}$ is a zero of multiplicity $\text{k}$ of a polynomial $\text{P}$ ”

Explanation of Solution

Given information:

The given statement is:

“What does it mean to say that $\text{c}$ is a zero of multiplicity $\text{k}$ of a polynomial $\text{P}$ ”

In the complete factorization theorem $\text{P}\left(\text{x}\right)\text{=a}\left(\text{x}-{\text{c}}_{\text{1}}\right)\left(\text{x}-{\text{c}}_{\text{2}}\right)\mathrm{...}\left(\text{x}-{\text{c}}_{\text{n}}\right)$

The numbers ${\text{c}}_{\text{1}}{\text{,c}}_{\text{2}}\mathrm{...}{\text{c}}_{\text{n}}$ are the zeroes of P. It is not necessary for all the zeroes to be different. If in the complete factorization of $\text{P}\left(\text{x}\right)$ the factor $\left(\text{x}-\text{c}\right)$ appears $\text{k}$ times, then it is said that $\text{c}$ is a zero of multiplicity $\text{k}$ of the polynomial $\text{P}$ .

d.

To determine

To state the zeroes theorem.

Explanation of Solution

Given information:

The given statement is:

“State the zeroes theorem”

The zeroes theorem states that in any polynomial of degree $\text{n}\ge 1$ , the polynomial has exactly $\text{n}$ zeroes, provided that a zero of multiplicity $\text{k}$ is counted $\text{k}$ times.

e.

To determine

To state the conjugate zeroes theorem.

Explanation of Solution

Given information:

The given statement is:

“State the Conjugate Zeroes Theorem”

The Conjugate Zeroes Theorem states that if a polynomial $\text{P}$ has real coefficients and if a complex number $\text{z}$ is a zero of $\text{P}$ then its complex conjugate $\overline{\text{z}}$ is also a zero of $\text{P}$ .