Three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. which statement describes the number and nature of all roots for this function? a. f(x) has two real roots and one imaginary root. b. f(x) has three real roots. c. f(x) has five real roots. d. f(x) has three real roots and two imaginary roots.

Step-by-step

Given:

Three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. Which statement describes the number and nature of all roots for this function?

A. f(x) has two real roots and one imaginary root.

B. f(x) has three real roots.

C. f(x) has five real roots.

D. f(x) has three real roots and two imaginary roots.

Solution:

We know from the fundamental theorem of algebra that every non-constant single variable polynomial with real coefficients of degree N has exactly N roots, multiplicities included.

Therefore there are five roots of in polynomial function f(x).

Of the five roots, we are already given that 2 of them are real, one of them is complex.

Since for a polynomial with real coefficients, each complex root (4+i), has its conjugate (4-i) as another root, so there is a minimum of 2 real and 2 complex roots.

The remaining (fifth) root must therefore be real, since we cannot have an odd number of complex roots.  This gives a total of three real and two complex roots.

1. sometimes

2.   never

3   .08c = 7.50+.05c   250 pages

step-by-step explanation:

1.   sometimes

the theoretical probability of rolling a 1 is 1/6.   perhaps you got really lucky and never rolled a 1   in those 60 times.   your experimental probability is 0.   or perhaps you rolled a 1   20 times.   your experimental probability is 20/60 is 1/3.     it really just depends on how you rolled in that experiment.

2.   never

the probability of the event occurring is 1.   that is being 100 % sure it will happen.   that is the maximum amount of confidence we have.

3.   no customer card cost:   .08 times number of pages

customer card cost:   7.50 + .05 times number of pages

we want to find when it costs the same amount, so we will set these equal

let c = number of pages

.08c = 7.50+.05c

subtract .05c from each side

.08c-.05c = 7.5+.05c-.05c

.03c=7.5

divide each side by .03

.03c/.03 = 7.5/03

c =250  

the cost is the same at 250 pages