Use Newton's Method to find the x-coordinate of the intersection of y = x⁴ + 2x³ and y = 2x² + 6x + 3. Use 3 for your initial value and calculate eight iterations. A) 2.3333333 B) 1.9347826 C) 1.7649117 D) 1.7331057 E) 1.7320508

Question

Quizplus · Accepted Answer

To find the x-coordinate of the intersection of $y = x^{14} + 2x^{13}$ and $y = 2x^{12} + 6x + 3$, we set them equal and solve for $x$. This gives us the equation $x^{14} + 2x^{13} - 2x^{12} - 6x - 3 = 0$. Using Newton's Method, where the next approximation is given by $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, and starting with $x_0 = 3$, after eight iterations, we approach the root of the equation. The correct answer, $1.7320508$, is the approximation of the root after these iterations, which is also the square root of 3, suggesting a special relationship in the polynomial's structure.

Use Newton's Method to Find the X-Coordinate of the Intersection