+20 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. 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 $f(x) + a\cdot g(x)$?
3. 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 $f(x) + b\cdot g(x)$?
4. Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. What is the degree of $f(x) \cdot g(x)$?
5. 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 $p(x) + q(x).$
+128448 1. 5x^3 + x^3 = 6x^3
2. a non-zero constant multiplier doesn't change thedegree of a polynomial
So the LARGEST possible degree of f(x) + a * g(x) = 4
+128448 3. Note that if b = -1/2 we have
(x^4 - 3x^2 + 2) - (1/2)(2x^4 -6x^2 + 2x - 1) =
(x^4 - 3x^2 + 2) - x^4 + 3x^2 - x + 1/2 =
-x + 2 + 1/2 =
- x + (5/2)x = degree 1
