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Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

You ask in a comment I am not sure, because if it was a real root, then I don't understand why this is used as an argument for introducing complex numbers? Having a real root feels unrelated to ...
Ethan Bolker's user avatar
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13 votes

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

I don't understand why a cubic always has at least one real root. I don't see why the reason is "since $y^3 − py − q$ is positive for sufficiently large positive $y$ and negative for ...
Alex Jones's user avatar
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11 votes

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

(this is just a simple historical explanation, without delving in the theory of cubic equations) The trouble with the solution of the cubic equation is that when the equation has three real solutions ...
mau's user avatar
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7 votes

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

First you were asking about: I don't see why the reason is "since $y^3 − py − q$ is positive for sufficiently large positive $y$ and negative for sufficiently large negative $y$)." Answer ...
Afntu's user avatar
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2 votes

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

With tools from Calculus the intermediate value theorem would do the trick. If calculus is not part of the tool box, the Descartes' sign change criteria works well for real polynomials of the form $x^...
2 votes
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What is a curve at $y=\infty$ mean?

It is explained to some extent in the link. Work in the projective plane, so introduce a variable $z$ and make every term in the polynomial homogeneous of degree 3: $$F(x,y,z) = xyz + ax^3 + bx^2z + ...
Ted Shifrin's user avatar
2 votes

Proof about intersection of three cubic curves

It is easy but first let us resolve a subtle issue. What if $R=S$? In that case we cannot conclude $L=L'$ (assuming $L'$ exists). We can easily bypass this by observing that given a fixed point $P$ ...
Ajin Shaji Jose's user avatar
1 vote

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

You already had good answers, so here is just a hint to grasp the idea: I don't see why the reason is "since $y^3−py−q$ is positive for sufficiently large positive $y$ and negative for ...
Basj's user avatar
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1 vote

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

A cubic function $y = Ax^3 + Bx^2 + Cx$ can have two turning values. If you plot it out, it can have a maximum and a minimum. Some cubic functions such as $y = x^3 + x$ have no turning values. A ...
Richard Kirk's user avatar
1 vote

Do the second differences of the fifth powers count the sphere packing of a polyhedron?

Those second differences are all multiples of $30$. Proof: $(n+1)^5-2n^5+(n-1)^5=20n^3+10n=20n(n+1)(n-1)+30n$ and $n(n+1)(n-1)$ must have a factor divisible by $3$. Divide out this factor of $30$ and ...
Oscar Lanzi's user avatar
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