Write a polynomial function of least degree with real coefficients in standard form that has -1, 3, and -i as zeros, and a leading coefficient of 1.

Question

Write a polynomial function in standard form of least degree that has real coefficients, -1,3, and -i as zeros, and a leading coefficient of 1.

Answer 1

Hi,

point of this question is if one of answer is -i you can be assure +i is also one of another zeros

x= -1. X=3. X= i. X= -i

f(x)= (x + 1) (x - 3) (x - i) (x + i)

f(x)= (x2 -2x -3)(x2 + 1)

f(x)= x4 -2x3 -2x2 -2x -3

NOTE: if one of zeros is a + bi, certainly you have a - bi as a other zeros ( at polynomial function with rational coefficient)

I hope it is useful,

Minoo

Answer 2

Hello Layla!

The real solutions are -1 and 3.

If 0 - i is a solution, the its conjugate is also a solution. Then 0 + i is also a solution.

We have then four solutions (i, -i, -1, 3)

So our polynomial can be written as:

(x + i)*(x - i)*(x + 1)*(x - 3) = (x² + 1)*(x² - 2x - 3) =

= x⁴ -2x³ - 3x² + x² - 2x -3 = x⁴ -2x³ - 2x² - 2x -3

Answer 3

P(x) = (x + 1)(x - 3)(x + i)(x - i)