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. | ||
| − | {{ | + | ==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 | ||
| + | |||
| + | <math>x^3 + ax^2 + bx + c = 0</math> | ||
| + | |||
| + | 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> | ||
| + | |||
| + | 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 23:55, 17 July 2018
A cubic polynomial is a polynomial of the form 
.
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
Now, we will make a change in variables to get rid of the 
term. If we do the substitution 
, this does the trick. Our new equation is of the form
We do another substitution 
, and our new equation is of the form
We can now turn this into a quadratic in terms of 
, solve for 
, and then solve for 
and finally 
.
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.
