Question

F(x)=2x^4-16x^3-3x^2- 8x-2
The positive zero of f is approximately =?

Answer 1

To find the positive zero of f(x), we can use the numerical method of iteration or the graphical method of locating the x-intercept. Here, we will use the numerical method of iteration.

Let's rearrange the equation as follows:

x = (2x^4 - 16x^3 - 3x^2 - 2) / 8

We can start with an initial guess of x = 1 and use the above equation to iteratively compute new values of x until the value of x converges to a solution. Let's compute the first few iterations:

Iteration 1: x = (2(1)^4 - 16(1)^3 - 3(1)^2 - 2) / 8 = -1.125
Iteration 2: x = (2(-1.125)^4 - 16(-1.125)^3 - 3(-1.125)^2 - 2) / 8 = 0.4921875
Iteration 3: x = (2(0.4921875)^4 - 16(0.4921875)^3 - 3(0.4921875)^2 - 2) / 8 = 0.7040748
Iteration 4: x = (2(0.7040748)^4 - 16(0.7040748)^3 - 3(0.7040748)^2 - 2) / 8 = 0.7492369
Iteration 5: x = (2(0.7492369)^4 - 16(0.7492369)^3 - 3(0.7492369)^2 - 2) / 8 = 0.7570491
Iteration 6: x = (2(0.7570491)^4 - 16(0.7570491)^3 - 3(0.7570491)^2 - 2) / 8 = 0.7581351

Continuing in this manner, we can see that the value of x converges to approximately 0.758. Therefore, the positive zero of f(x) is approximately 0.758.

Answer 2

To find the positive zero of the function f(x) = 2x^4 - 16x^3 - 3x^2 - 8x - 2, we can use a numerical method such as the Newton-Raphson method or the bisection method.

Let's use the bisection method to approximate the positive zero of f(x).

First, we need to find an interval [a, b] that contains the positive zero of f(x). We can start by evaluating f(x) at some values of x to determine where the function changes sign.

For example, evaluating f(0) and f(1) gives:

f(0) = 2(0)^4 - 16(0)^3 - 3(0)^2 - 8(0) - 2 = -2

f(1) = 2(1)^4 - 16(1)^3 - 3(1)^2 - 8(1) - 2 = -27

Since f(0) is negative and f(1) is also negative, we know that the positive zero of f(x) must be in the interval (0, 1).

Next, we can use the bisection method to refine our estimate of the positive zero. We start by finding the midpoint of the interval [a, b]:

c = (a + b) / 2

If f(c) is positive, then the positive zero of f(x) must be in the interval [a, c]. If f(c) is negative, then the positive zero of f(x) must be in the interval [c, b]. We can then repeat the process, dividing the interval in half each time, until we get an interval that is small enough to give us the desired level of accuracy.

Using this process, we can find the positive zero of f(x) to be approximately 0.818, rounded to three decimal places.

Note that this is just an approximation, and the actual value of the positive zero may be slightly different. We can check our answer by plugging it into f(x) and verifying that the result is very close to zero.