If `2x=3+5i,` then what is the value of `2x^(3)+2x^(2)-7x+72`?

Options

A. 4
B. `-4`
C. 8
D. `-8`

Answer

A

Explanation

Given `2x=3+5i`
`impliesx=(3+5i)/(2)`
Consider `x^(3)=(27+125i^(3)+225i^(2)+135i)/(8)`
`=(27-125i-225+135i)/(8)" "({:(becausei^(2),=-1),(i^(3),=-i):})`
`=(-198+10i)/(8)=(-99+5i)/(4)`
`and x^(2)=(9+25+30i)/(4)`
`=(9-25+30i)/(4)=(-8+15i)/(2)`
Now, Consider `2x^(3)+2x^(2)-7x+72`
`=((-99+5i)/(2))+(-8+15i)-(7(3+5i))/(2)+72`
`=-(99)/(2)+(5i)/(2)-8+15i-(21)/(2)-(35)/(2)i+72`
`=(-(99)/(2)-8-(21)/(2)+72)+((5)/(2)+15-(35)/(2))i`
`=(-99-16-21+144)/(2)=8/2=4`