now, this polynomial has roots of 3-i and 4i, namely 3 - i and 0 + 4i.
let's bear in mind that a complex root never comes all by her lonesome, her sibling is always with her, the conjugate, so if 3 - i is there, 3 + i is also coming along, likewise if 0 + 4i is there, her sibling 0 - 4i is also there.
To solve the question, we will be using a Venn Diagram. Let's list down what's given first:
855 drank alcohol
657 smoked cigarettes
192 used illegal drugs
418 both drank alcohol and smoked cigarettes
109 drank alcohol and used illegal drugs
116 smoked cigarettes and used illegal drugs
88 engaged in all three
and 251 engaged in none of these
We will draw a Venn Diagram with 3 circles, one for each vice. Then we will start with the intersection of all three and the area outside of the circles (for those who do not engage in vices).
Because we know how many engaged in all 3 vices, we can deduct that number from those who belong to the intersection of 2 vices since they have already been counted in that too.
418 - 88 = 330 drank and smoked, but not used drugs
109 - 88 = 21 drank and used drugs, but not smoked
116 - 88 = 28 smoked and used drugs but not drank
Then, we subtract the numbers in each circle from the total for each vice.
855 - (330 + 88 + 21) = 416 drank only
657 - (330 + 88 + 28) = 211 smoked only
192 - (88 + 21 + 28) = 55 used drugs only
To find the total number of surveyed students, we just need to add all the numbers in the Venn Diagram.
416 + 330 + 88 + 21 + 211 + 28 + 55 + 251 = 1,400
Therefore, there were 1,400 students who were surveyed.
