Solve the Problem $3 \mathrm { x } ^ { 4 } + 2 \mathrm { x } - 4 = 0$

question 295

Essay

Solve the problem.
-Use Newton's method to estimate the solutions of the equation $3 \mathrm { x } ^ { 4 } + 2 \mathrm { x } - 4 = 0$ . Start with $\mathrm { x } _ { 1 } = 1$ for the right-hand solution and with $x _ { 0 } = - 1.5$ for the solution on the left. Then, in each case find $x _ { 2 }$ .

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Definitions:

Method Of Least Squares

A mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets ("the residuals") of the points from the curve.

Squared Deviations

The squared differences between individual values and the mean of a dataset, used in statistical analysis to measure variance.

Scatter Diagram

A visual depiction employing points to illustrate the connection between two numerical variables.

Cov(X,Y)

The covariance between two variables X and Y, measuring how changes in one variable are associated with changes in the other.