I will give 50 points
The function f(x) is a cubic function and a limited table of values is provided below. Write the equation of the cubic polynomial in standard form.

f(x) = x³+2x²-16x-32

Step-by-step explanation:

f(x) = 0 when x = -4, -2, 4

So these are the roots

Factors are (x+4)(x+2)(x-4)

f(x) = [(x+4)(x-4)](x+2)

f(x) = (x²-16)(x+2)

f(x) = x³+2x²-16x-32

f(x) = 1x^3 + 2x^2  -16x - 32

Step-by-step explanation:

Looking at the first row of the table, I see f(x) = -27 when x = -5.  We know that -27 is the cube of -3.  So, try adding 2 to x = -5, obtaining -3.  The result (-5 + 2)^3 equals -27, as we wanted.

Unfortunately the proposed function f(x) = (x + 2)^3 does not "work" for other x values.

The governing polynomial involves four coefficients:  f(x) = ax^3 + bx^2 + cx + d.

We can pick any four x values and the corresponding y-values and come up with four versions of the polynomial f(x) = ax^3 + bx^2 + cx + d:

(1)  Choose x = -4 and f(x) = 0 from the second line in the table.  Then

     0 = a(-4)^3 + b(-4)^2 + c(-4) + d, or -64a + 16b - 4c + d = 0, and

(2) Choose x = 0 and y  = -32.  Then -32 = f(x) = a(0)^3 + b(0)^2 + c(0) + d.  Then we have -32 = 0 + 0 + 0 + d, or d = -32.

(3) Choose x = 4 and y = 0.  Then we have 0 = 64a + 16b + 4c - 32.

(4) Choose x = 5 and y = 63.  Then 63 = 125a + 25b + 5c - 32

Now we have four equations in four unknowns {a, b, c, d}.

Let's simplify these four equations, as follows:

(2) We have already seen that d = -32.  Therefore,

(3) becomes 64a + 16b + 4c = 32

(4) becomes 125a + 25b + 5c = 95.  We need one more equation.

(5) will be based on (2, -48):  -48 = 2^3a + 2^2b + 2c - 32, or

                                                 -16 = 8a + 4b + 2c

Now we have a system of linear equations to solve:

  8a + 4b + 2c = -16

125a +25b+5c = 95

 64a  + 16b + 4c = 32

Here I used the MATRIX functions of my old TI-83Plus calculator to find the values of {a, b, c}.  They are:

a=1, b=2 and c= -16

and so the general 3rd order polynomial becomes

f(x) = 1x^3 + 2x^2  -16x - 32

for finding the midpoint of any given 2 pair of points simply add the respective x coordinates with other x coordinate and the y coordinate with other y coordinate and divide them by 2

in your question it will be :

(3+1)/2 for the x coordinate of mid point i.e 2

[8+(-4)]/2 for the y coordinate of midpoint i.e 2

so coordinates of mid point of points (3,8) and (1,-4) will be (2,2)

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation: