Answer:
Positive roots = 1 or 0
Negative roots = 0, 2, or 4
Complex roots = 0, 2, 4, or 6
Step-by-step explanation:
We are given the polynomial:
f(x) = -x⁶ - x⁵ - x⁴ - 4x³ − 12x² + 12
Now, by inspection, the highest degree is 6 and as such the polynomial is therefore a polynomial with more than 2 as a degree.
Applying Descartes Rule of Signs to the polynomial :
f(x) = -x⁶ - x⁵ - x⁴ - 4x³ − 12x² + 12
Signs are: - - - - - +
There is only 1 sign change and thus, it means there is 1 or 0 positive roots
To find number of negative roots, we will use f(-x);
f(−x) = -(−x)⁶ - (−x)⁵ - (−x)⁴ - 4(−x)³ − 12(−x)² + 12(−x)
f(-x) = -x⁶ + x⁵ - x⁴ + 4x³ − 12x² - 12
Signs are: - + - + - -
There are 4 sign changes and thus, it means there are 4, 2 or 0 negative roots
Thus; Complex roots = 0, 2, 4, or 6
