Solve the following equation in the complex number system
x^4+x^3+4x^2+10x-60=0
hint
x = 2 and x = -3 are roots
so either do long division by x-2 and x+3
to get a quadratic or do it by synthetic division.
Solve that quadratic using the formula.
so what would my equation be to figure this question out?
To solve the given equation in the complex number system, we can use various methods such as factoring, completing the square, or using the quadratic formula. However, the given equation is a quartic equation, which means it is of degree 4. Solving quartic equations can be quite involved by hand. In this case, we will use a numerical method called Newton's method to approximate the roots.
Newton's method is an iterative numerical method that can find the roots of a function by starting with an initial guess and successively refining it until a desired level of accuracy is achieved.
To apply Newton's method to solve the given equation, we need to first rewrite it as a function f(x) equal to zero. Then we can find the roots by following these steps:
Step 1: Express the equation as a function f(x) equal to zero:
f(x) = x^4 + x^3 + 4x^2 + 10x - 60 = 0
Step 2: Calculate the derivative of the function f'(x).
f'(x) = 4x^3 + 3x^2 + 8x + 10
Step 3: Choose an initial guess for x_0. This initial guess can be any complex number. Let's start with x_0 = 1.
Step 4: Iterate using Newton's method until the desired level of accuracy is achieved:
Repeat the following steps until the desired accuracy or number of iterations is reached:
- Calculate the next iterate x_{n+1} using the formula:
x_{n+1} = x_n - f(x_n) / f'(x_n)
- Check if the absolute value of the difference between x_{n+1} and x_n is less than the desired accuracy. If it is, stop the iteration.
- Otherwise, set x_n = x_{n+1} and go back to the previous step.
Step 5: Repeat Step 4 for different initial guesses to find all the roots of the equation.
Keep in mind that since the equation is a quartic equation, it can have up to four complex roots. Since it is computationally intensive to follow Newton's method by hand, it is more practical to use numerical software or programming to solve it.