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 $f(x) + g(x)$?
The x^3 coefficient will have a sum of 1 + 5 = 6
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 $f(x) + a•g(x)$?
Depends upon the value of "a"
Note that if a = -1/2, the sum of the first two terms in each polynomial is 0....and the sum of the second two terms in each polynomial is also 0....so the sum will produce a degree of 1.....any other value of "a" will produce a 4th degree polynomial
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 $f(x) + b•g(x)$?
Just like the last, if "b" is -1/2, the smallest possible degree will be 1
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 $f$?
Guess that we have a third degree polynomial of the form ax^3 + bx^2 + cx + d
If f(0) = 47, then d = 47
And we have this system
a + b + c + 47 = 32
8a + 4b + 2c + 47 = -13
27a + 9b + 3c + 47 = 16
The solution to this is a = 52/3, b = -67, c = 104/3, d = 47
And the sum of these is 32
