1. 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 \(f(x) + a\cdot g(x)\)?
2. 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 \(f(x) + b\cdot g(x)\)?
+6251 1)
\(a\cdot g(x) \text{ will always have degree 4 for }a\neq 0\\ \text{assuming no cancellation the sum will have degree 4 as well}\\ \text{this is the largest degree the sum can have}\)
2)
\(\text{let }b=-\dfrac 1 2\\ f(x)+b\cdot g(x) = (1-1)x^4 + (-3+3)x^2 -x+\left(2+\dfrac 1 2\right)=\\ -x+\dfrac 5 2 \text{ which has degree 1}\)
