+1243 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. Suppose $f$ is a polynomial such that $f(0) = 47$, $f(1) = 32$, $f(2) = -13$, and $f(3)=16$. What is the sum of the coefficients of $f$?
5.Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. What is the degree of $f(x) \cdot g(x)$?
6.Find $t$ if the expansion of the product of $x^3 - 4x^2 + 2x - 5$ and $x^2 + tx - 7$ has no $x^2$ term.
7. There is a polynomial which, when multiplied by $x^2 + 2x + 3$, gives $2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9$. What is that polynomial? (Give your answer as a polynomial in which the terms appear in order of decreasing degree. In other words, "$x^5 + x + 2$" is a valid answer, but "$2+x^5+x$" is not.)
8. What is the coefficient of $x$ in $(x^4 + x^3 + x^2 + x + 1)^4$?
9. What is the coefficient of $x^3$ in this expression? (x^4 + x^3 + x^2 + x + 1)^4
