1. ## Polynomial Funtion HELP PLEASE!!!

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).

2. Originally Posted by rlarach

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).

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

3. Originally Posted by rlarach

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).

$\displaystyle f(x)=x^2+6x-12$

RonL

4. How Do You Get X^2 + 6x - 12

5. Originally Posted by rlarach
How Do You Get X^2 + 6x - 12
coefficient 1, so is of the form:

f(x)=x^2+bx+c.

Then:

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

Now expend the middle term and equate coefficients.

RonL