Polynomial Inequalities: "Suppose a sonar device is set up in the middle of a lagoon.

noahpww

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Polynomial Inequalities: "Suppose a sonar device is set up in the middle of a lagoon.

Hello Again,


I have another 3-part problem that I could use some assistance with.


2a. Suppose a sonar device is set-up in the middle of a lagoon. The device found the depth of the lagoon in feet, as a function of the horizontal distance, in yards, from the device is given by D(x) = (x² − 4)³ − 5(x² − 4)² + (x² − 4) − 5. Factor the formula for D completely. Do not solve for anything.


I tried factoring this and got D(x) = (x + 3)(x - 3)(x4 − 8x2 + 17) Is this correct?


2b. Draw and complete a number line for the inequality D(x) > 0.


How would I got about creating a number line for this?


2c. Assuming the lagoon is circular, using the information from your number line in part (b), what is the diameter of the pond?


(A side not for this entire 3-part problem, D(x)=0 is at the surface of the lagoon. Assume the positive direction to be "upwards.")
 
Hello Again,


I have another 3-part problem that I could use some assistance with.


2a. Suppose a sonar device is set-up in the middle of a lagoon. The device found the depth of the lagoon in feet, as a function of the horizontal distance, in yards, from the device is given by D(x) = (x² − 4)³ − 5(x² − 4)² + (x² − 4) − 5. Factor the formula for D completely. Do not solve for anything.


I tried factoring this and got D(x) = (x + 3)(x - 3)(x4 − 8x2 + 17) Is this correct?
Yes, it is.


2b. Draw and complete a number line for the inequality D(x) > 0.


How would I got about creating a number line for this?
Draw a line! Mark 0 and the positive and negative integers on the line so that each point can be identified with a real number. (Surely you have seen "number lines" before?). Mark the zeroes of this polynomial on that line. x= 3 and x= -3 are two obvious ones. If we let \(\displaystyle z= x^2\) we can write that last factor as \(\displaystyle z^2- 8z+ 17\). By the quadratic formula its zeroes are \(\displaystyle \frac{8\pm\sqrt{64- 68}}{2}= 4\pm 2i\). Those are not real so \(\displaystyle z^2- 8z+ 17\) has no real zeroes. The only real zeroes of the original polynomial are 3 and -3.

They divide the number line into 3 intervals: x< -3, -3< x< 3, and x> 3. Every polynomial is "continuous" so -3 and 3 are the only places the polynomial can change sign. Calculate the polynomials value at one point in each interval to determine whether D(x) is positive or negative on that interval.

2c. Assuming the lagoon is circular, using the information from your number line in part (b), what is the diameter of the pond?


(A side note for this entire 3-part problem, D(x)=0 is at the surface of the lagoon. Assume the positive direction to be "upwards.")

Since "D(x)= 0 at the surface of the lagoon" and we are to "assume the positive direction to be "upwards"", the lagoon itself is the interval where D(x) is negative. What is the length of that interval?
 
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