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 $f(x) + b\cdot g(x)$?
If f(x) = x4 - 3x2 + 2 and g(x) = 2x4 - 6x2 + 2x - 1.
If you create the function c·g(x) and let c = -½ then c·g(x) = -½( 2x4 - 6x2 + 2x - 1 )
= - x4 + 3x2 - x + ½
f(x) + c·g(x) now is (x4 - 3x2 + 2) + (- x4 + 3x2 - x + ½) = 2 - x + ½ = -x + 2½
and has degree 1.