Final answer:
The P-intercepts of the function P(r) = 3r⁴ + 21r³ +30r² are (0, 0), (-2, 0), and (-5, 0).
Step-by-step explanation:
The P-intercepts of a function are the points at which the function crosses or intersects the x-axis. To find the P-intercepts, we set P(r) equal to zero and solve for r.
Given the function P(r) = 3r⁴ + 21r³ +30r², we can set it equal to zero: 3r⁴ + 21r³ +30r² = 0.
Now we can factor out the common term r² to simplify the equation: r²(3r² + 21r + 30) = 0.
The equation is equal to zero when either r² = 0 or 3r² + 21r + 30 = 0.
From r² = 0, we find r = 0. Therefore, the point (0, 0) is a P-intercept of the function.
From 3r² + 21r + 30 = 0, we find r = -2 and r = -5. Therefore, the points (-2, 0) and (-5, 0) are also P-intercepts of the function.
Therefore, the answer is: A) (0, 0), B) (-2, 0), and C) (-5, 0) are all P-intercepts of the function.