1. Let $f(x) = x^3 + 3x ^2 + 4x - 7$ and $g(x) = -7x^4 + 5x^3 +x^2 - 7$. What is the coefficient of $x^3$ in the sum $f(x) + g(x)$? 2(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 $f(x) + a\cdot g(x)$?
2(b). 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 $f(x) + b\cdot g(x)$?
3. Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. What is the degree of $f(x) \cdot g(x)$?
4. The degree of the polynomial $p(x)$ is 11, and the degree of the polynomial $q(x)$ is 7. Find all possible degrees of the polynomial $p(x) + q(x).$