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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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2 Answers

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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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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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