Quadratics of quadratics

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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.
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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.
 
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  • #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.

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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.

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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.

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