Let f(x) be a polynomial function such that, for all real x, f(x²+2)=x^4+10x²+4. Evaluate f(x²-2)?

Question

Let f(x) be a polynomial function such that, for all real x,

f(x²+2)=x^4+10x²+4.

Evaluate f(x²-2)

This is a prec-calculus question that i need help with....please someone help me....i dont understand it!!! :s

Answer 1

f(x^2+ 2) = x^4 + 10x^2 + 4

= x^4 + 4x^2 + (2)^2 + 6x^2

=(x^2 + 2)^2 + 6x^2

=(x^2 + 2)^2 + 6x^2 + 12 - 12

f(x^2+ 2) = (x^2 + 2)^2 + 6(x^2 +2) - 12

put x^2 + 2 = y

f(y) = y^2 + 6y - 12

when x^2 + 2 = y

x^2 = y -2

x^2 - 2 = y -4

f(x^2 - 2) = f(y-4) = (y-4)^2 + 6(y-4) - 12

y^2 + 16 - 8y + 6y - 24 - 12

= y^2 - 2y -20

substitute back y = x^2 + 2

f(x^2-2) = (x^2+2)^2 - 2(x^2+2) - 20

= x^4 + 4x^2 + 4 - 2x^2 - 4 - 20

= x^4 + 2x^2 - 20

Answer 2

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Answer 3

(x - square root of 2) (x + square root of 2)

This is for the last one.