 In this video, we provide the solution to question number five for practice exam number four for math 1210 We're given the function f of x which equals x cube minus x squared minus 4x minus 2 And we're asked to apply Newton's method given the initial value x 1 equals 2 and we're supposed to find x 2 in that sequence So this would be a sequence that approximate an x intercept our root of this polynomial function here So we have to use the formula x 2 is going to equal x 1 minus f of x 1 Over f prime at x 1 so we do do need to know the derivative This is a polynomial function so the derivatives not so bad the derivative will be 3x squared minus 2x minus 4 And so x 1 is given as 2 so we need to plug those into Newton's formula here So we're going to get 2 minus evaluate the function at 2 the original function So we get 2 cube which is 8 minus 2 squared, which is 4 minus 4 times 2, which is 8 Minus 2 then we have to evaluate this at the derivative as well So we plug in 2 2 squared is 4 times 3 is 12 Minus 2 times 2 which is 4 minus 4 So let's try to simplify this thing from here So looking in the numerator notice there's an 8 minus 8 so those get each other And so we get negative 4 minus 2 that's going to be a negative 6 in the numerator and the denominator You're gonna get 12 minus 4 minus 4 So tell 12 to your weight before excuse me 12 take away 4 is 8 take away another 4 is gonna be for itself And so let's see you have a double negative. So it's gonna be 2 plus 6 over 4 But 6 and 4 we could reduce the fraction as 3 halves. So to add those together We think of 2 actually as 4 halves plus 3 halves We see that the approximation is gonna be 7 halves which then gives us that x 2 is C