Points of Intersection

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paperangel

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Sep 14, 2005
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Directions:
Find all points of intersection for the fuctions:

f(x) = 9x^6 + 56x^4 + 12x^3 - 120x^2 - 70x + 30
g(x) = 16x^6 - x^5 + 6x^3 - 13x^2 -15x + 20


I've tried to solve it graphically which was an absolute nightmare - so is it possible to do it algebraically or is that impossible? I think I can do it if I just know how I am supposed to do it.
 
Where they intersect, they're equal, so:

. . . . .9x<sup>6</sup> + 56x<sup>4</sup> + 12x<sup>3</sup> - 120x<sup>2</sup> - 70x + 30 = 16x<sup>6</sup> - x<sup>5</sup> + 6x<sup>3</sup> - 13x<sup>2</sup> - 15x + 20

. . . . .0 = 7x<sup>6</sup> - x<sup>5</sup> - 56x<sup>4</sup> - 6x<sup>3</sup> + 107x<sup>2</sup> + 55x - 10

Then apply the usual polynomial tools: the Rational Roots Test, synthetic division, etc, etc. I'll give you a start: x = 1/7 is a solution.

Eliz.
 
That makes so much more sense now. Thank you very much. You are a hero. :p :p :p I'm glad you can put up with all of this non-sense. You truely are a blessing.
 
I've been working this problem for the past two hours and I'm stumped:

Rational roots test: ± 1, 2, 5, 10, 1/7, 2/7, 5/7, 10/7

So far I was told that 1/7 is a solution, and have found that 2, and -1 are also.

I've found that -1/7, ±2/7, ±10/7, -2, ±5/7, and 1 are not solutions.

This has left me with:

7x^4 + 14x^3 - 28x^2 - 70x - 35

But where do I go from here? Do I factor it?
 
If you've found three zeroes, then you should be down to a cubic. Why do you still have a quartic polynomial?

Also, have you verified that none of the listed zeroes occurs twice? Because if one does occur twice, then you would be able to get down to a quadratic, at which point the Quadratic Formula applies. (This is how this sort of exercise usually works, is why I ask.)

Eliz.
 
Wow. I'm embarrassed that I didn't think to check if any of the answers appeared more than once. :oops: :oops: :oops: Negative one went in twice more. You are so good to me. If I knew who you were, I'd make you cheesecake.
 
paperangel said:
Wow. I'm embarrassed that I didn't think to check if any of the answers appeared more than once.
Click to expand...
No biggie. This was just one of those things that's "obvious" when you're old and boring and have done this stuff for twenty years. :oops: :wink: :lol:

Glad I could help!

Eliz.
 
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