Polynomial calculations Find a polynomial f that satisfies the following properties. Determine the degree of f. then substitute a polynomial of that degree and solve for its coefficients. f(f(x))=9x−8 f(f(x))=x4−12x2+30

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Polynomial calculations Find a polynomial f that satisfies the following properties. Determine the degree of f. then substitute a polynomial of that degree and solve for its coefficients.  f(f(x))=9x−8  f(f(x))=x4−12x2+30

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Step 1 1) Given that, f(f(x))=9x−8...(1) Step 2 Since degree of given composition is 1 so degree of f is also 1. Consider, f(x)=ax+b So (f⋅f)(x)=f(f(x)) =f(ax+b) =a(ax+b)+b(Since f(x)=ax+b) =a2x+ab+b...(2) Step 3 Using (1) and (2), 9x−8=a2x+ab+b Compare the coefficient of x and constant term, a2=9⇒a=±3 And ab+b=−8⇒b=−8a+1 When a=3,b=−2 When a=−3,b=4 So When a=3 and b=−2 then f(x)=3x−2 When a=−3 and b=4 then f(x)=−3x+4 Step 4 Hence, f(x)=3x−2 f(x)=−3x+4

Polynomial calculations Find a polynomial f that satisfies the following properties. Determine the degree of f. then substitute a polynomial of that degree and solve for its coefficients. f(f(x))=9x-8 f(f(x))=x^{4}-12x^{2}+30

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