Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

4, -8, and 2 + 5i

Answer 1

use the fact that an n'th degree polynomial with roots \(x=r_1,x=r_2,...,x=r_n\) can be written as:\[f(x)=C(x−r_1)(x−r_2)...(x−r_n)\]where C is some constant.

in addition, if a polynomial has any complex roots, then they ALWAYS appear in pairs where each pair contains a complex number and its complex conjugate.

example: if a polynomial has roots 4 and 1+2i, then we can write it as:\[f(x)=C(x−4)(x−(1+2i))(x-(1-2i))\]since (1-2i) is the complex conjugate of (1+2i)

Answer 2

would this be right f(x) = x4 - 6x3 + 20x2 - 122x + 928

Answer 3

no, it is not.
Following what @asnaseer told you, you should multiply
\[ \large (x-4)(x+8)(x-(2+5i))(x-(2-5i)) \]

Answer 4

f(x) = x4 - 19x2 + 244x - 928

Answer 5

yes, well done

Answer 6

a tip for complex roots:
first write it as
x = 2 +- 5i
then get the imaginary part alone
x-2 = +-5i
square both sides
(x-2)^2 = 25i^2
simplify
x^2 -4x +4 = -25
set equal to zero
x^2 -4x +29 = 0

--> (x-4)(x+8)(x^2-4x+29)

Answer 7

nice tip @dumbcow :)

Answer 8

and well done @babydoll332 :)

Answer 9

thank you

Answer 10

yw :)

Answer 11

thank you guys as well :D