Answer:
a) H(-x) = 4x⁴ + 5x³ +2x² + x + 5
b) 0 sign changes
c) 0 negative real solutions
d) 4 imaginary solutions
Explanation:
a) Replace every x with (-x) and rewrite the function below
H(x) = 4x⁴ - 5x³ + 2x² - x + 5
H(-x) = 4(-x)⁴ - 5(-x)³ + 2(-x)² - (-x) + 5
H(-x) = 4x⁴ + 5x³ +2x² + x + 5
b) How many sign changes are there in this new function?
0 (zero) sign changes in the new function (terms 5x³ and x)
c) How many negative real roots are possible?
0 (zero) negative real solutions possible.
d) How many possible complex roots are there?
There are 4 or 2 or 0 positive roots for the function and 4 total roots for this function; Given the degree of the function is also 4, there can be 0 or 2 or 4 complex roots for the function. If you were to graph H(x), you would see that the function never touches or crosses the x-axis. Therefore, the function has 4 imaginary roots and no real roots.