Quadratics of quadratics

  • Thread starter okkvlt
  • Start date
In summary, the conversation discusses the possibility of expressing a polynomial of degree 2^n as n-many quadratic functions of quadratic functions. However, it is not possible to find a unique solution for the coefficients in this scenario, as there are more equations than unknowns.
  • #1
okkvlt
53
0
Question about quadratics of quadratics

can any 4th degree polynomial be expressed as a quadratic of a quadratic function?


or in the more general case, can any polynomial of degree 2^n be expressed as n-many quadratic functions of quadratic functions?
and given a polynomial of degree 2^n is there a way to find the coefficients of the quadratics?


i tried finding a way to reduce the 4th degree poly into a quadratic of a quadratic. but the main problem is when trying to find the coefficients that i have 5 equations in 6 unknowns, so the system to find the coefficients isn't determined. why and what does this mean?


r=(R4)X^4+(R3)X^3+(R2)X^2+(R1)X+R0

p[q[X]]=P2(Q2*X^2+Q1*X+Q0)^2+P1(Q2*X^2+Q1*X+Q0)+P0

in order for r[x] to equal p[q[x]]:

R4=(P2)(Q2)
R3=2(P2)(Q2)(Q1)
R2=(P2)(2(Q2)(Q0)+(Q1)^2)+(P1)(Q2)
R1=2(P2)(Q0)(Q1)+(P1)(Q1)
R0=(P2)(Q0)^2+(P1)(Q0)+(P0)

im wondering if i could just set one of the coefficients of either quadratic equal to zero. but even then the system is unsolveable by matrices, and I am not sure what terms are allowed to be zero.
 
Mathematics news on Phys.org
  • #2
? Yes...? Are you really asking about composition of functions? If so, then, yes.
 
  • #3
No you can't.

If it was possible then you could always take Q0=0 and Q2=1, just by absorbing these constants into p. Then, you have 5 equations in 4 unknowns.
 

What are "Quadratics of Quadratics"?

"Quadratics of Quadratics" refers to the study and analysis of quadratic equations that involve other quadratic equations as their variables. This means that the coefficients and constants in the equation are themselves quadratic expressions.

How are "Quadratics of Quadratics" different from regular quadratic equations?

While regular quadratic equations involve only one quadratic variable, "Quadratics of Quadratics" involve multiple quadratic variables. This adds an extra layer of complexity and can result in more than one solution for the equation.

What are some real-world applications of "Quadratics of Quadratics"?

"Quadratics of Quadratics" can be used to model complex systems in fields such as physics, engineering, and economics. They can also be used in optimization problems, where the goal is to find the maximum or minimum value of a function.

How do you solve "Quadratics of Quadratics" equations?

To solve "Quadratics of Quadratics" equations, you can use a variety of methods such as substitution, elimination, or graphing. In some cases, it may also be necessary to use advanced techniques such as completing the square or the quadratic formula.

What are some common mistakes to avoid when working with "Quadratics of Quadratics"?

One common mistake is not properly distributing the quadratic variables before solving the equation. It is also important to carefully check the signs and coefficients when substituting values into the equation. Additionally, it is important to check for extraneous solutions, which may arise due to the multiple layers of variables in "Quadratics of Quadratics" equations.

Similar threads

Replies
4
Views
736
  • General Math
Replies
16
Views
3K
Replies
2
Views
720
Replies
5
Views
3K
  • Mechanical Engineering
Replies
3
Views
1K
  • Programming and Computer Science
Replies
4
Views
826
  • Introductory Physics Homework Help
Replies
2
Views
649
Replies
5
Views
3K
  • Linear and Abstract Algebra
Replies
4
Views
2K
  • General Math
Replies
3
Views
4K
Back
Top