How are cubic equations converted into this form?

  • Context: Undergrad 
  • Thread starter Thread starter autodidude
  • Start date Start date
  • Tags Tags
    Cubic Form
Click For Summary
SUMMARY

Cubic equations can be expressed in the form y = a(x-b)^3 + c, a process known as "completing the cube." However, not all cubic equations can be transformed into this format. For instance, the equation x³ + 4x² + 3x + 1 can be rewritten as (x+1)³ + x², but this requires specific conditions on the coefficients. Unlike completing the square, which has a standard method, completing the cube lacks a universally accepted approach due to the necessity of correctly related coefficients for successful transformation.

PREREQUISITES
  • Understanding of cubic equations and their properties
  • Familiarity with polynomial manipulation techniques
  • Knowledge of completing the square method
  • Basic algebraic skills for handling coefficients
NEXT STEPS
  • Research the method of completing the cube in detail
  • Study polynomial long division for cubic equations
  • Explore the relationship between coefficients in cubic equations
  • Learn about transformations of polynomial functions
USEFUL FOR

Students studying algebra, mathematicians interested in polynomial transformations, and educators teaching cubic equations and their manipulations.

autodidude
Messages
332
Reaction score
0
y = a(x-b)^3 + c

I'm not sure what it's called, my book doesn't mention on how it's derived.
 
Mathematics news on Phys.org
Well, not all cubic equations can be converted into this form. For example, x3+4x2+3x+1 can only be converted into (x+1)3+x2, and unless you let c be non-constant, it can't be put into the form you're looking for.
 
I guess it would be called "completing the cube" like "completing the square".

But to be able to "complete the square" you must be given coefficients of [itex]x^2[/itex] and [itex]x[/itex] while to "complete the cube" you would have to be given coefficients of [itex]x^3[/itex], [itex]x^2[/itex], and x. And, as Char.Limit said, unlike completing the square, if those coefficients are not related correctly, it will be impossible. Which is probably why there is not standard name for the operation!
 

Similar threads

Undergrad Quadratic, cubic, quartic, quintic equations
  • · Replies 16 ·
Replies
16
Views
5K
Undergrad Does this make a cubic out of a quadratic system?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Why was the substitution x = y - b/3a used in Cardano's Cubic Formula?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad The solution to a cubic equation
  • · Replies 2 ·
Replies
2
Views
2K
MHB Understanding Cubic Equation Formula for Polynomials of Degree Three
  • · Replies 9 ·
Replies
9
Views
4K
MHB Solving Cubic Equation: x^3 - kx + (k + 11) = 0
  • · Replies 3 ·
Replies
3
Views
2K
High School Problem of solving the cubic function
  • · Replies 4 ·
Replies
4
Views
2K
MHB Using Substitution to Solve Cubic Equations
  • · Replies 3 ·
Replies
3
Views
2K
MHB How Long to Empty Lake Erie with a Cone Cup?
  • · Replies 2 ·
Replies
2
Views
2K
High School Normal form of the conic section equation
  • · Replies 11 ·
Replies
11
Views
8K