Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b\cdot g(x)$?
The smallest possible degree is 1.
Let b = (-1/2)
x^4 - 3x^2 + 2 + (-1/2) [ 2x^4 - 6x^2 + 2x - 1 ] =
x^4 - 3x^2 + 2 - x^4 + 3x^2 - x + 1 =
-x + 3
The smallest possible degree ( as found by the guest) = 1