Hello there. To solve this question, we'll need to remember some properties about polynomial functions.
Say we have f(x) = ax² + bx + c and g(x) = dx² + ex + h, for a, b, c, d, e, h constants and a, d not equal to zero.
Defining the sum (f + g)(x) = f(x) + g(x), we'll have to analyze each following case:
A) Degree of the sum
There are two possibilities. First, a and d have different absolute values, so for example they cannot cancel each other, getting rid of the second degree term of the function.
If a + d is not equal to zero, then the function will still be of degree two.
If a + d is equal to zero, the function can either be of the first or zero degree (a constant)
B) How many zero roots
We'll need to actually sum the functions to analyze the case:
(f + g)(x) = (a + d)x² + (b + e)x + (c + h)
The roots of this polynomial can be found by using the quadratic formula
(f + g)(x) = 0 for x = (-(b + e) + - sqrt((b + e)² - 4 (a + d) (c + h)))/(2(a + d))
If the expression in the square root is greater than zero, the function will have two distinct real roots.
If it is equal to zero, the function will have two equal real roots.
If it is less than zero, the function will not have real roots (they'll be two conjugate complex roots)
C) Let's say we have the functions f(x) = x² + x + 1 and g(x) = 2x² + 3x + 4
The sum function is (f + g)(x) = 3x² + 4x + 5
Plugging in those coefficients on the expression in the square root, we'll have:
4² - 4*3*5
16 - 60
- 44
As you can see, in this case, the expression will give us a negative number, which means that the sum function will have no real roots.