Determine a quadratic function f(x) that satisfies this equation:
f(x)*f(x-1)= f(x^2)
f(x) = Ax^2 + Bx + C
f(x-1) = A(x-1)^2 + B(x-1) + C
= Ax^2 + (-2A+B)x + (A-B+C)
f(x^2) = Ax^4 + Bx^2 + C
f(x)*f(x-1) = (Ax^2 + Bx + C)*(Ax^2 + (B-2A)x + (A-B+C))
= A^2x^4 + (AB-2A^2+AB)x^3 + (A^2-AB+AC + B^2-2AB + AC)x^2 + (AB-B^2+CB+CB-2AC)x + (AC-BC+C^2)
= A^2x^4 + 2A(B-A)x^3 + (A^2-3AB+2AC+B^2)x^2 + (AB-B^2+2CB-2AC)x + (AC-BC+C^2)
A^2 = A so A=1, since A=0 would mean: not quadratic
2A(B-A) = 0 so B=A=1
A^2-3AB+2AC+B^2 = B (let A=B=1)
1 - 3 + 2C + 1 = B = 1
so C=1
A=B=C=1
f(x) = x^2 + x + 1
f(x-1) = x^2 - 2x + 1 + x - 1 + 1 = x^2 - x + 1
f(x^2) = x^4 + x^2 + 1
(x^2 + x + 1) * (x^2 - x + 1) = x^4 + x^2 + 1 checks
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Posted by Larry
on 2024-02-02 10:43:32 |
solution
