let f be a polynomial such that f(x^2+1)=x^4+4x^2. determine f(x^2-1)
To determine the value of f(x^2 - 1), we will first find an expression for f(x^2 + 1) and then substitute x^2 - 1 in place of x^2 in that expression.
From the given equation: f(x^2 + 1) = x^4 + 4x^2
Let's first substitute x^2 = t in the equation to obtain f(t + 1) = t^2 + 4t.
Now, let's find an expression for f(t) using the substitution t = x^2 + 1.
Since x^2 = t - 1, we can write the equation as f(t) = (t - 1)^2 + 4(t - 1)
Expanding this expression, we get f(t) = t^2 - 2t + 1 + 4t - 4
Simplifying further, we have f(t) = t^2 + 2t - 3
Now, we can substitute x^2 - 1 in place of t to find f(x^2 - 1):
f(x^2 - 1) = (x^2 - 1)^2 + 2(x^2 - 1) - 3
Expanding and simplifying:
f(x^2 - 1) = (x^4 - 2x^2 + 1) + 2x^2 - 2 - 3
= x^4 + 1
Therefore, f(x^2 - 1) = x^4 + 1.