Let \(g(x) = x^2 - 11x + 30.\) Find the polynomial \(f(x)\) with positive leading coefficient, that satisfies \(g(f(x)) = x^4 - 14x^3 + 62x^2 - 91x + 42.\)
+128407 g ( x) = x^2 - 11x + 30
Let's assume that f(x) is of the form x^2 + bx + c
Then
g (f(x) = [ x^2 + bx + c ] ^2 - 11 [ x^2 + bx + c ] + 30 = x^4 -14x^3 + 62x^2 -91x + 42
Simplify
x^4 + 2bx^3 + (b^2 + 2c) x^2 + 2bcx + c^2 - 11x^2 - 11bx -11c + 30 =
x^4 -14x^3 + 62x^2 -91x + 42
x^4 + 2bx^3 + (2c + b^2 - 11)x^2 + ( 2bc - 11b)x + [c^2 -11c + 30] =
x^4 - 14x^3 + 62x^2 - 91x + 42
Equate coefficients
2b = -14
So b = -7
2c + b^2 - 11 = 62
2c + 49 - 11 = 62
2c + 38 = 62
2c = 24
c = 12
So
f (x) = x^2 - 7x + 12
Check
g(f(x) =
[ x^2 - 7x + 12 ] ^2 - 11 [ x^2 - 7x + 12 ] + 30 =
x^4 - 14 x^3 + 62 x^2 - 91 x + 42
