Hello, I wasn't sure if this question belonged here or under another section. In the grade 12 math text I'm using, I've learned pretty much 3 main ways to solve quadratic equations: 1) Synthetic Division, 2) the Factor Theorem, and 3) the Quadratic Formula. The problem I have I will show in a few steps:
1) 3t^3 - t^2 - 6t + 2 = 0
The roots are: 1/3; and + or - (square root of 2)
2) 2m^3 - 5m^2 + 1 = 0
The roots are: 1/2; and 1 + or - (square root of 2)
So far, the main approach I use is the factor theorem for quadratics with higher than a power of 2, so I can get the roots and determine the factors. For example:
x^4 - 27x = 0
R=P(3)=(3)^4 - 27(3)
Therefore, 3 is a root, meaning (x-3) is a factor.
The problem I have is using this process I just showed you for the examples 1) and 2) above. Basically, I use factors of the # with the smallest power of x: i.e. 27x, I would test 1,-1, 3, -3, 9, -9, 27, -27.
The problem is that I don't have a good method of determining the roots for 1) and 2), since they are fractions and/or square roots, and since I have to substitute them into x^2 and x^3, it's a bit difficult for me to see the answer right away. Is there a fairly simple method to finding these that anyone could show me? Strangely enough, out of over 200 questions this chapter, these are the only two like this, and I can't solve them because I can't find a factor to work with in order to continue to the next steps.
Thanks for your time, sorry about the lengthy post.
Edit: oops, minor mistake