How do you find the solutions to the following equation?
x^4-x^3+x-1=0
First, identify what kind of solution we are searching for via Des Cartes Rule of Sign.
3 or 1 positive
1 negative
2 or 0 imaginary
I don't know how to format synthetic division in latex so here is the best I have at it.
If , our division looks like:
Hence x=1 is a solution.
3 or 1 positive
1 negative
2 or 0 imaginary
We found 1 positive solution and we know there is 1 negative solution. We have 2 choices of solutions .
If we take our new polynomial, , we can do synthetic division again to obtain a quadratic polynomial.
If , our division looks like:
Now we have
We can now use the quadratic equation, , to solve the quadratic.