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I need some leads and guidance on my homework question:

Find a cubic equation of the form, $y = ax^3 + bx^2 +cx + d$ and a straight line equation $y = mx + k$ (m is non-zero) such that the straight line intersects the cubic three times.

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Did you draw a picture? –  draks ... Mar 26 '12 at 10:53
    
You can easily guarantee that they have one point (with a particularly-nice $x$ coordinate) in common. From there, can you see how to guarantee two more? (I have a different, much better hint, but it gives too much away.) –  Blue Mar 26 '12 at 12:11

2 Answers 2

HINT Write down a polymonial with roots at $x_0$, $x_1$ and $x_2$.

$y=(x-x_0)(x-x_1)(x-x_2)$

Assume that this is $(ax^3 + bx^2 +cx + d)-(mx+k)$.

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+1 though "assume" looks a little strange here –  Henry Mar 26 '12 at 11:30
    
@Henry Assuming is great. It helps out a lot when dealing with this problems. –  Pedro Tamaroff Mar 26 '12 at 19:21
    
I would have said something like "choose $a,b,c,d,m,k$ so that the two expressions are identical" –  Henry Mar 26 '12 at 21:49

Here's a more explicit hint than what draks gave. Pick any three numbers to be the $x$-coordinates of the intersection, say $1,$ $2,$ $3.$ Write down a cubic equation that has these three numbers as solutions. One way to do this is by giving the equation in factored form so that it's obvious the three numbers are solutions, such as $(x-1)(x-2)(x-3)=0.$ Now pick any line to use in the intersection, say $y = 3x - 2.$ Finally, think about what you get if you add $3x - 2$ to both sides of the cubic equation two sentences back.

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