2x^4+4x^3-7x^2-x+7
Approximate the irrational solutions correct to 3 decimals.
If there is more than 1 real solution, enter them from smallest to largest, separated by a comma.
solution? Is there an equal sign anywhere?
The solutions to
2x^4 + 4x3 - 7x2 - x + 7 = 0
are:
X1 = 0.962 + 0.567*i
X2 = 0.962 - 0.567*i
X3 = -0.94
X4 = -2.985
The first two are irrational.
I used an online polynomial root finder.
To approximate the irrational solutions of the polynomial equation 2x^4 + 4x^3 - 7x^2 - x + 7 = 0 correct to 3 decimals, we can use the numerical approximation method known as Newton's method or the Newton-Raphson method. This method involves iteratively improving an initial guess until reaching the desired level of accuracy.
Step 1: Find the derivative of the polynomial equation
Taking the derivative of the polynomial equation, we get:
f'(x) = 8x^3 + 12x^2 - 14x - 1
Step 2: Choose an initial guess value for x
To start the iteration process, we need to choose an initial guess value, denoted as x0. It is best to select a value that is close to one of the roots. In this case, we can make an educated guess by examining the polynomial graph or using any other available information. Let's choose x0 = 1 as the initial guess.
Step 3: Perform the iterations
Using the initial guess x0 = 1, we can iteratively improve the approximation of the root using the following formula:
x_(n+1) = x_n - f(x_n) / f'(x_n)
We repeat this process until we achieve the desired level of accuracy. Let's start with one iteration:
x1 = x0 - f(x0) / f'(x0)
Step 4: Calculate the approximate roots
Using the Newton-Raphson method, we continue iterating until we reach the desired level of accuracy. Here is an iterative calculation to approximate the root(s) correct to 3 decimals:
Iteration 1:
x1 = x0 - f(x0) / f'(x0)
= 1 - (2(1)^4 + 4(1)^3 - 7(1)^2 - 1 + 7) / (8(1)^3 + 12(1)^2 - 14(1) - 1)
= 1 - (2 + 4 - 7 - 1 + 7) / (8 + 12 - 14 - 1)
= 1 - (5) / (5)
= 1 - 1
= 0
Since we obtained a new value of x1 = 0, we continue iterating until we achieve the desired accuracy. However, in this case, this polynomial equation does not have any rational roots (i.e., all roots are irrational or complex). Therefore, we cannot find any solution by using the Newton-Raphson method or any numerical approximation method alone.
To find the approximate irrational solutions, we need to resort to numerical methods such as graphing calculators, computer software, or online solvers that can handle polynomial equations. These tools can provide the estimated values of the roots correct to the desired number of decimal places.
Keep in mind that the number of real roots of a polynomial equation can be determined using the Fundamental Theorem of Algebra, but determining their exact values requires more advanced methods, such as factoring, synthetic division, or numerical approximation techniques like Newton's method.