Hi
How do I show that 2 + 8x - 2x^3 = 0 has three real solutions?
I know you can say it has at most three solutions from the fundamental theorem of algebra, and I presume the easiest way to show the three solutions are real and not complex is to actually find them... But how ought one to go about this?
I can't see any convenient factoring up the equation..
Moderator edit: This thread has been moved from the Pre-Calculus to the Calculus subforum. This comment is made so that some of the posts (which I choose not to delete) in this thread make sense. For the record, if a question requires differentiation or integration then it is clearly a calculus question and belongs in the Calculus subforum.
You know more than this: any real polynomial of odd degree has at least one real zero.
Well, maybe you are allowed to use a little calculus: Determine the high and low point (if any) of the graph ofBut how ought one to go about this?
I can't see any convenient factoring up the equation..
You will see that there is a high point above the x-axis and a low point below the x-axis. - Therefore, from the intermediate value theorem, we can conclude that there must be three real zeros of f.
Ah, but this is the Pre-calculus forum, so we cannot use this method. Either that, or the OP should have posted this in the Calculus subforum.
So without calculus, you'd best graph the function. If you really want to solve it algebraically, I guess you could try the cubic formula (see here: Cubic function - Wikipedia, the free encyclopedia) but I wouldn't recommend it. It's messy.
01
That's the thing. There is no set definition of what Precalculus is, so I don't fault you for posting in the wrong forum. Where I live:
Precalculus =
advanced algebra (many schools call this "College Algebra") plus
trigonometry (and yes, we have a separate trig subforum) plus
analytic geometry (conic sections, polar coordinates, vectors) plus
a little linear algebra (systems of linear equations and matrices) plus
a little discrete math (binomial theorem, sequences & series, mathematical induction).
Your mileage may vary.
01
you can use the Intermediate Value Theorem -- from Wolfram MathWorld