Solve the following equation in the complex number system
x^4+x^3+4x^2+10x-60=0
College Algebra - Reiny, Monday, January 2, 2012 at 11:49pm
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.
College Algebra - Jennifer, Tuesday, January 3, 2012 at 8:10pm
so what would my equation be to figure this question out?
Do what Reiny said, step by step
****__x^3__+3x^2__+10x__+30________
x-2 | x^4 + x^3 + 4 x^2 + 10 x - 60
******x^4 -2x^3
******----------------------------
**********3 x^3 + 4 x^2 + 10 x - 60
**********3 x^3 - 6 x^2
**********--------------------------
****************+10 x^2 + 10 x -60
****************+10 x^2 - 20 x
******************-----------------
*************************+30 x -60
*************************+30 x -60
*************************---------
Remainder = 0
Now divide x^3+3x^2+10x+30
by x + 3
That will leave you with a quadratic which you solve with the quadratic equation, probably resulting in two complex conjugate roots.
Find X such that the point(x,9) is 10 units from (9,3)
To solve the equation x^4 + x^3 + 4x^2 + 10x - 60 = 0, you can follow the hint given by the previous response. The hint suggests that x = 2 and x = -3 are roots of the equation.
To find the equation that needs to be solved, you can divide the given equation by (x-2)(x+3). This can be done using long division or synthetic division.
Let's use synthetic division to divide the equation by (x-2)(x+3):
-3 | 1 1 4 10 -60
| -3 6 -30
---------------------
| 1 -2 10 -20
The quotient of the division is 1x^2 - 2x + 10 and the remainder is -20.
So, the equation that needs to be solved is: (x-2)(x+3)(x^2 - 2x + 10) = 0
Now, we have factored the original equation into two factors: (x-2)(x+3) and (x^2 - 2x + 10) = 0.
To solve the quadratic factor (x^2 - 2x + 10) = 0, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac))/(2a)
Comparing the quadratic factor (x^2 - 2x + 10) = 0 with the quadratic formula, we can see that a = 1, b = -2, and c = 10.
Applying the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(1)(10)))/(2(1))
Simplifying this expression gives us:
x = (2 ± √(4 - 40))/2
= (2 ± √(-36))/2
Since we are solving the equation in the complex number system, we can use the imaginary unit "i" to represent the square root of -1.
Therefore, the solution to the equation x^4 + x^3 + 4x^2 + 10x - 60 = 0 in the complex number system is:
x = 2, -3, 1 + 3i, 1 - 3i