Question

Part a: Let f(x)=x^4-3x^2+2 and
g(x)=2x^4-6x^2+2x-1. Let a be a constant. What is the largest possible degree of f(x)+a*g(x)?

Part b: Let 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*g(x)?

Answer 1

(a)
f and g are both degree 4, so their sum cannot be of higher degree.

(b)
1/2 g(x) = x^4-3x^2+x - 1/2

f - 1/2 g = -x + 5/2

Looks like 1 is the smallest possible degree.