Find all solutions of the equation.
x^4 + 3x^3 − 13x^2 − 9x + 30 = 0
the Rational Root Theorem shows that the roots are among
±1,±2,±3,±5,±6,±10,±15,±30
A little synthetic division shows that
x^4 + 3x^3 − 13x^2 − 9x + 30 = (x-2)(x^3+5x^2-3x-15)
A little inspection via grouping gives you
(x-2)(x+5)(x^2-3)
so, ...
Try x = 2
16+24-52-18+30 = 0 :) so (x-2) is a factor
or if all else fails
https://www.mathportal.org/calculators/polynomials-solvers/polynomial-factoring-calculator.php
(x-2)(x^2-3)(x+5)
2 , +sqrt 3 , -sqrt 3 , -5
To find all the solutions of the equation x^4 + 3x^3 - 13x^2 - 9x + 30 = 0, you can use different methods such as factoring, using the rational root theorem, or numerical methods.
Let's try factoring first. Unfortunately, factoring a quartic equation like this one is often quite challenging. It's unlikely that you'll be able to find a simple factored form for this equation. So, we'll need to use an alternative method.
A common approach to solving quartic equations is to factorize them into quadratics. We can start by making a substitution to simplify the equation.
Let's substitute y = x^2. This simplifies the equation to:
y^2 + 3y - 13(x^2 + 9/13)x + 30 = 0
Now, we can treat y as the variable and try to factorize this quadratic equation. However, this may not always be possible, especially for more complex equations.
Alternatively, we can use numerical methods to find the approximate solutions. One such method is the Newton-Raphson method, which iteratively approaches the solutions. However, implementing this method can be mathematically involved.
Another method is to use a graphing calculator or a computer software to graph the equation. By observing the graph, you can estimate the approximate values of the solutions.
In this case, I recommend using a computer software or a graphing calculator to graph the equation and find the approximate solutions.