Final answer:
The intervals in which the polynomial 2x² − x − 5 is negative or positive can be determined by finding its roots and testing values in the surrounding intervals. The polynomial is negative between the roots and positive outside of them.
Step-by-step explanation:
To determine the intervals where the polynomial 2x² − x − 5 is positive or negative, we first need to find the roots of the equation by setting it equal to zero. Solving 2x² − x − 5 = 0 will provide the x-values that are the boundaries for the intervals of interest. The quadratic formula, x = ∛(-b ± √(b² - 4ac))/(2a), is applied here, with coefficients a = 2, b = -1, and c = -5.
Calculation of the discriminant, b² - 4ac, gives (-1)² - 4(2)(-5) = 1 + 40 = 41, which is positive, indicating two real and distinct roots. The roots are calculated as follows:
- x = (1 + √41)/4
- x = (1 - √41)/4
Once we have the roots, we can test values in each interval to determine where the function is positive or negative. Since the leading coefficient of the polynomial is positive (2), the parabola opens upwards, and the polynomial will be negative between the roots and positive outside of them.
The intervals where the polynomial is negative are between the two roots, and the intervals where it is positive are to the left of the smaller root and to the right of the larger root.