Help!! The roots of a function are at x=-1, x=1/4i, and x=-1/4i. Which statement is true about this function?
A. No such function exists
B. The graph of the function crosses the x-axis only one
C. The graph of the function is increasing over any interval.
D. The expanded form of the function will have an imaginary coefficient.

Question

Help!! The roots of a function are at x=-1, x=1/4i, and x=-1/4i. Which statement is true about this function?
A. No such function exists
B. The graph of the function crosses the x-axis only one
C. The graph of the function is increasing over any interval.
D. The expanded form of the function will have an imaginary coefficient.

zehanz · Answer

Well, A is wrong, because you can define:$$f(x)=(x+1)(x-\frac{ 1 }{ 4 }i)(x+\frac{ 1 }{ 4 }i)$$so such a function does exist.

What do you think, B, C or D as answer?

anonymous · Answer

D?

zehanz · Answer

How many times do you think the graph crosses the x-axis?

anonymous · Answer

Once..?

zehanz · Answer

It must, because -1 is a zero. It also must be the only time, otherwise there would be more real zeros! B is the right answer.

BTW if you multiply out the formula of f, you get:$$f(x)=(x+1)(x^2 -\frac{ 1 }{ 16 }i^2)=(x+1)(x^2+\frac{ 1 }{ 16 }) = ...$$See? There are no more real zeros, also there is no imaginary coefficient.

anonymous · Answer

Thank you so much! Sorry for being such a hassle!!

zehanz · Answer

No problem!