Let f(x) be a polynomial function such that, for all real x.
f(x^2 +2) = x^4 + 10x^2 + 4.
Evaluate f(x^2 - 2).
THANKS FOR YOUR HELP!!!!!!
Well, the "brute force" method would be:
$\displaystyle f(x^2 + 2) = x^4 + 10x^2 + 4$
Let $\displaystyle y = x^2 + 2$.
Solve for x:
$\displaystyle x^2 = y - 2$
$\displaystyle x = \pm \sqrt{y - 2}$
Ignoring the difficulty of the $\displaystyle \pm$ let's blithely continue:
$\displaystyle f(y) = ( \pm \sqrt{y - 2} )^4 + 10 ( \pm \sqrt{y - 2} ) + 4$
$\displaystyle f(y) = (y - 2)^2 + 10(y - 2) + 4$
Now let $\displaystyle y = x^2 - 2$.
$\displaystyle f(x^2 - 2) = (x^2 - 2 - 2)^2 + 10(x^2 - 2 - 2) + 4$
$\displaystyle f(x^2 - 2) = (x^2 - 4)^2 + 10(x^2 - 4) + 4$
which you can expand, or not, as you choose.
-Dan