1) what are the real zeroes of x3 + 6 x2 – 9x – 54?
a. 1, 2, 27
b. 3, -3, -6
c. -6, 3, -6
d. 2, -1, 18
e. 3, 3, -6
2) an entrepreneur estimates that the total profit (profit = total revenue minus total cost) from his proposed company will be given by the function p(x) = x3 – 4x2 + 3x – 12,
where p is in hundreds of dollars and x is the number of years elapsed after the company starts operations. in how many years (x) will the company break even
(no profit, no loss)?
a. 1 year
b. 2 years
c. 3 years
d. 4 years
e. 6 years
3)how many negative roots can this polynomial function possibly have?
f(x) = 5x4 – 7x3 – x2 – 3x – 2
a. 1
b. 5 or 3 or 1
c. 3 or 1
d. 4 or 2 or 0
e. 2 or 0

Question 1)

The polynomial function is:

It could be written as:

The roots of a function are the possible values of x where f(x)=0

i.e.

Since,

Hence, we get:

Hence, the real zeros of the given problem is:

       B.   3, -3, -6

Question 2)

The profit function is given by:

We are asked to find the point where the company break even i.e. there is no profit or loss i.e. the possible value of x such that P(x)=0

i.e.

i.e.

( As the square of a number can't be negative.

Hence, is not a solution)

Hence, the solution is:

The correct answer is:

      D.  4 years    

Question 3)

The polynomial function is given by:

We know that the number of sign changes of function f(x) gives the possible number of positive real roots

and  the number of sign changes of function f(-x) gives the possible number of negative real roots.

The number of sign changes in f(x) is 1

(Hence, f(x) has at most 1 positive real root)

and

The number of sign changes in f(x) is 3

(Hence, f(x) has at most 3 negative real root)

              The correct answer is:

                 C.      3 or 1

17   x 256=4352

sorry if this is wrong, i'm new.

step-by-step explanation:

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

=====================================

part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

=====================================

part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).