Difference between revisions of "Cubic polynomial"

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A '''cubic polynomial''' is a [[polynomial]] of the form <math>ax^3+bx^2+cx+d=0</math>.
 
A '''cubic polynomial''' is a [[polynomial]] of the form <math>ax^3+bx^2+cx+d=0</math>.
 
A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form . An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.
 
A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form . An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.
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==Solving a cubic==
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If the leading coefficient of the cubic is not 1, then divide both sides by the leading coefficient so it is 1. You get an equation of the form
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<math>x^3 + ax^2 + bx + c = 0</math>
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Now, we will make a change in variables to get rid of the <math>x^2</math> term. If we do the substitution <math>x = y - \frac{a}{3}</math>, this does the trick. Our new equation is of the form
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<math>y^3 + py = q</math>
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We do another substitution <math>y = w - \frac{p}{3w}</math>, and our new equation is of the form
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<math>w^3 + rw^{-3} = q</math>
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We can now turn this into a quadratic in terms of <math>w^3</math>, solve for <math>w</math>, and then solve for <math>y</math> and finally <math>x</math>.
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Because cubic polynomials have an odd degree, their end behaviors go in opposite directions, and therefore all cubic polynomials must have at least one real root.

Revision as of 00:55, 18 July 2018

A cubic polynomial is a polynomial of the form $ax^3+bx^2+cx+d=0$. A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form . An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.

Solving a cubic

If the leading coefficient of the cubic is not 1, then divide both sides by the leading coefficient so it is 1. You get an equation of the form

$x^3 + ax^2 + bx + c = 0$

Now, we will make a change in variables to get rid of the $x^2$ term. If we do the substitution $x = y - \frac{a}{3}$, this does the trick. Our new equation is of the form

$y^3 + py = q$

We do another substitution $y = w - \frac{p}{3w}$, and our new equation is of the form

$w^3 + rw^{-3} = q$

We can now turn this into a quadratic in terms of $w^3$, solve for $w$, and then solve for $y$ and finally $x$.

Because cubic polynomials have an odd degree, their end behaviors go in opposite directions, and therefore all cubic polynomials must have at least one real root.