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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.