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 \(f(x) + a\cdot g(x)\)?
I assume you may have meant to say smallest, as I have seen those kinds of questions before, so I will answer both.
For largest, because a is a constant, it cannot make the largest exponent any higher, so the answer is 4.
For smallest, we want to cancel some exponents, so we can multiply g(x) by -1/2 and add to get x^4 - 3x^2 + 2 - x^4 + 3x^2 - x + 1/2 = -x + 5/2, so the degree is 1.
Hoping this helped,