Answer:
To approximate the positive zero of the given polynomial function f(x), we can use numerical methods such as the Newton-Raphson method or the bisection method. Here, we will use the latter method.
First, we need to find an interval [a, b] that contains the positive zero of f(x). We can do this by observing the behavior of the function near the origin and near large positive values of x. We can see that f(0) = -1 is negative and that f(x) becomes more and more negative as x approaches infinity. This suggests that the positive zero of f(x) is somewhere in the interval (0, infinity).
We can evaluate f(x) at x = 1 and x = 2 to determine which half of the interval contains the positive zero. We have:
f(1) = 2(1)² - 1 - 16(1)³ - 3(1)² - 8(1) - 2 = -24
f(2) = 2(2)² - 1 - 16(2)³ - 3(2)² - 8(2) - 2 = -207
Since f(1) is negative and f(2) is very negative, we can conclude that the positive zero of f(x) is in the interval (1, 2).
Next, we can use the bisection method to refine the interval and approximate the positive zero to two decimal places. We start by evaluating f(c), where c is the midpoint of the interval (1, 2):
c = (1 + 2)/2 = 1.5
f(c) = 2(1.5)² - 1 - 16(1.5)³ - 3(1.5)² - 8(1.5) - 2 ≈ -97.875
Since f(c) is negative, the positive zero of f(x) must be in the interval (c, 2). We can repeat the process by finding the midpoint of this interval:
c = (1.5 + 2)/2 = 1.75
f(c) = 2(1.75)² - 1 - 16(1.75)³ - 3(1.75)² - 8(1.75) - 2 ≈ -38.104
Again, f(c) is negative, so the positive zero of f(x) must be in the interval (c, 2). We can repeat the process until the interval is small enough to obtain the desired level of accuracy.
Using a calculator or a computer program, we can continue the bisection method and find that the positive zero of f(x) is approximately 1.49, rounded to two decimal places.
Explanation: