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Let \(g(x) = x^2 - 11x + 30.\) Find the polynomial \(f(x)\) with positive leading coefficient, that satisfies \(g(f(x)) = x^4 - 14x^3 + 62x^2 - 91x + 42.\)

 Mar 9, 2022
 #1
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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

 

 

cool cool cool

 Mar 9, 2022

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