Chapter 4.1, Problem 31E

(a)

To determine

To find: whether graph represents s a polynomial function and then state the number of complex zeros and the number of real zeros.

Answer to Problem 31E

It has 1 real root, i.e. $x=-a$

It has 1 complex root as: $-a=-a+0i$

Explanation of Solution

Given information: The graph

  

Calculation:

Since the graph (which is at least) a 3^rd degree polynomial,

Crosses the x-axis once,

It has 1 real root, at, for example $x=-a$ .

Therefore it also has 1 complex root as:

  $-a=-a+0i$

It has 1 real root, i.e. $x=-a$

It has 1 complex root as: $-a=-a+0i$

(b)

To determine

graph represents a polynomial function and state the number of complex zeros and the number of real zeros of each function.

Answer to Problem 31E

It has 2 real roots, i.e. $x=\pm a$

It has 2 complex roots as: $\pm a=-a+0i\text{ and }a+0i$

Explanation of Solution

Given information: The graph

  

Calculation:

Since the graph (which is at least) a 2^nd degree polynomial,

Crosses the x-axis twice,

It has 2 real roots, at, for example $x=\pm a$ .

Therefore it also has 2 complex roots since:

  $\pm a=-a+0i\text{ and }a+0i$

It has 2 real roots, i.e. $x=\pm a$

It has 2 complex roots as: $\pm a=-a+0i\text{ and }a+0i$

(c)

To determine

To find: graph represents a polynomial function and state the number of complex zeros and the number of real zeros of each function.

Answer to Problem 31E

It has 2 real roots, i.e. $x=-a\text{ and }-b$

It has 2 complex roots as: $-a=-a+0i\text{ and }-b=-b+0i$

Explanation of Solution

Given information: The graph

  

Calculation:

Since the graph (which is at least) a 4^th degree polynomial,

Crosses the x-axis twice,

It has 2 real roots, at, for example $x=-a\text{ and }-b$ .

Therefore it also has 2 complex roots since:

  $-a=-a+0i\text{ and }-b=-b+0i$

It has 2 real roots, i.e. $x=-a\text{ and }-b$

It has 2 complex roots as: $-a=-a+0i\text{ and }-b=-b+0i$