Form a​ third-degree polynomial function with real​ coefficients, with leading coefficient​ 1, such that 6+i and 5 are zeros.

A third-degree polynomial can be written as:

f(x) = a*x^3 + b*x^2 + c*x + d

Where the leading coefficient is a, and all the coefficients are real.

If we know that the leading coefficient is 1, then the equation becomes:

f(x) = x^3 + b*x^2 + c*x + d

Now, we also know that:

(6 + i) and 5 are zeros.

This means that:

(6 + i)^3 + b*(6 + i)^2 + c*(6 + i) + d = 0

remember that:

i^2 = - 1

This is equal to:

(6 + i)*(36 + 2*6*i  + i^2) + b*(36 + 2*6*i  + i^2) + c*(6 + i) + d = 0

(6 + i)*(35 + 12i) + b*(35 + 12i) + c*(6 + i) + d  =0

(210 + 35i + 72i - 12) +  b*(35 + 12i) + c*(6 + i) + d  = 0

198 + 107i +  b*(35 + 12i) + c*(6 + i) + d  = 0

sparating in real and imaginary part, we get:

(198 + b*35 + c*6 + d) + (107 + b*12 + c)*i = 0

Then each parentheses needs to be zero, this means that:

198 + b*35 + c*6 + d = 0

107 + b*12 + c =  0

Knowing that 5 is another zero, we have:

5^3 + b*5^2 + c*5 + d = 0

125 + b*25 + c*5 + d = 0

Then we have a system of 3 equations and 3 variables:

198 + b*35 + c*6 + d = 0

107 + b*12 + c =  0

125 + b*25 + c*5 + d = 0

To solve this, we first need to isolate one of the variables in one of the equations.

Let's isolate d in the last one, so we get:

d = -125 - b*25 - c*5

now we can replace this in the first equation to get:

198 + b*35 + c*6 + d = 0

198 + b*35 + c*6 + (  -125 - b*25 - c*5) = 0

70 + b*10 + c = 0

So now we have two equations:

70 + b*10 + c = 0

107 + b*12 + c =  0

Again, now we can isolate the one variable in one of the equations, this time let's isolate c in the first one.

c = -70 - b*10

now we can replace this in the other equation:

107 + b*12 + c = 0

107 + b*12 + (-70 - b*10) = 0

38 + b*2 = 0

now we can solve this for b

b*2 = -38

b = -38/2 = -19

Now, with the equation c = -70 - b*10 we can find the value of c.

c = -70 - b*10 = c = -70 - (-19)*10 = 120

And with the equation  d = -125 - b*25 - c*5

we can find the value of d:

d = -125  - b*25 - c*5 = -125 - (-19)*25 - (120)*5 = -250

Then we have:

a = 1

b = -19

c = 120

d = -250

The eqation is:

f(x) = 1*x^3 - 19*x^2 + 120*x - 250

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.

walter needs to make at least 5 more cookies but no more than 40. 5 ≤ x ≤ 40

step-by-step explanation:

20 ≤ x + 15 ≤ 55

subtract 15 from all parts

20-15 ≤ x + 15-15 ≤ 55-15

5 ≤ x ≤ 40

this means how many more cookies he has to make

walter needs to make at least 5 more cookies, but no more than 40 cookies; 5 ≤ x ≤ 40