solve the equation in the complex number system
x^4+x^3+4x^2+10x-60=0
To solve the given equation in the complex number system, we can use the factoring method combined with synthetic division.
Step 1: Factor the polynomial as much as possible.
We can attempt to factor the equation by applying the Rational Root Theorem. According to the theorem, any rational root of the equation, if it exists, must be of the form ±p/q, where p is a factor of the constant term (-60) and q is a factor of the leading coefficient (1).
Considering the factors of 60, the possible rational roots can be ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, and ±60.
To check for the roots, we will perform synthetic division for each possible rational root until we find one that results in a remainder of 0.
Step 2: Perform synthetic division.
By performing synthetic division with the possible roots, we find that x = 2 is a root with a remainder of 0.
2 | 1 1 4 10 -60
-2 -2 4 28
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1 -1 2 14 -32
The quotient obtained after synthetic division is 1x^3 - 1x^2 + 2x + 14, and the remainder is -32.
Therefore, (x - 2) is a factor of the polynomial, and we can rewrite the equation as:
(x - 2)(x^3 - x^2 + 2x + 14) = 0
Step 3: Solve the remaining cubic equation.
Now, we need to solve the cubic equation represented by (x^3 - x^2 + 2x + 14) = 0. To do this, we can use other algebraic methods such as factoring, the rational root theorem, or numerical methods like Newton's method.
Unfortunately, the given cubic equation does not factor easily and does not have any rational roots. Therefore, we can use numerical methods, such as graphing calculators or computer software, to approximate the solutions. These methods will provide complex solutions for the remaining equation.
Hence, the complete set of solutions to the given equation in the complex number system is x = 2 along with the complex solutions obtained from solving the cubic equation (x^3 - x^2 + 2x + 14) = 0 using numerical methods.