Tangents that pass through the origin

[Hyacinth42]

Homework Statement

At how many points on the curve 4x$$^{}5$$ - 3x$$^{}4$$ + 15x$$^{}2$$ + 6 will the line tangent to the curve pass through the origin?

Homework Equations

The Attempt at a Solution

I have no idea of how to even approach this... Erm, I fiddled around with the point-slope formula of a line

y = mx - mx1 + y1

where (x1, y1) is the point on the line/curve and m is the slope

and in order for the line to pass through the origin, then mx1 + y1 have to cancel out... But I have no idea of how to find a place on the line where this occurs... Am I going about this all wrong/is there an easier way to do it? If not, where do I go from here?

 

[HallsofIvy]

I will assume that you mean "the curve y= $4x^5- 3x^4+ 15x^2+ 6$" If x= $x_1$ then y= $40x_1^5- 3x_1^4+ 15x_1^2+ 6$ and m=y'= $20x_1^4- 12x_1^3+ 20x_1$. Now use that "mx1+ y1= 0".

 

[Dick]

You are on the right track. So for the tangent line m=y1/x1. And the slope of the tangent line is dy/dx. So dy/dx=y/x. If you put the y polynomial into that you will have problems actually solving for x. The polynomial degree is 5 and it doesn't factor. On the other hand, the question asks you just to count the roots. I'm not sure I really know an elegant way to do that, except by messing around with it's derivatives to figure out where it's increasing and decreasing etc.