Answer:
(i) p = -7, q = -3
(ii) (-3/2, 0), (2, 0), (3, 0)
Step-by-step explanation:
Use long division (see picture).
When we divide by x − 1, we get a remainder of q + p + 20.
When we divide by x + 1, we get a remainder of 16 − q + p.
We know the first remainder is equal to 10 and the second remainder is equal to 12. Solving the system of equations:
10 = q + p + 20
12 = 16 − q + p
0 = q + p + 10
0 = 4 − q + p
0 = 14 + 2p
p = -7
q = -3
Therefore:
f(x) = 2x³ − 7x² − 3x + 18
Finding the zeros:
f(x) = 2x³ − 6x² − x² − 3x + 18
f(x) = 2x² (x − 3) − (x² + 3x − 18)
f(x) = 2x² (x − 3) − (x − 3) (x + 6)
f(x) = (x − 3) (2x² − (x + 6))
f(x) = (x − 3) (2x² − x − 6)
f(x) = (x − 3) (x − 2) (2x + 3)
The zeros are at x = -3/2, x = 2, and x = 3.
