Question

Write a function with the given characteristics: a polynomial with rational coefficients having roots 3, 3, and 3-i

Answer 1

@Vocaloid

Answer 2

3-i is one root
the conjugate 3+i is another root

they come in pairs

Answer 3

x = 3-i or x = 3+i
x-3 = -i or x-3 = i
(x-3)^2 = (-i)^2 or (x-3)^2 = (i)^2
(x-3)^2 = -1

Answer 4

if you were to solve (x-3)^2 = -1 for x, you'd get x = 3-i or x = 3+i

Answer 5

(x-3)^2 = -1

(x-3)^2 + 1 = 0

x^2 - 6x + 9 + 1 = 0

x^2 - 6x + 10 = 0


so if 3-i is a root, then x^2 - 6x + 10 is a factor

Answer 6

if 3 is a root, then x-3 is a factor
this is a double root, so (x-3)^2 is a factor

Answer 7

x^2 - 6x + 10 is a factor
(x-3)^2 is a factor

so you'll have (x-3)^2*(x^2-6x+10)

expand that all out to get the final answer

Answer 8

what do you mean by expand that out?

Answer 9

@jim_thompson5910

Answer 10

hopefully you see how (x-3)^2 turns into x^2 - 6x + 9 ?

Answer 11

you get that by multiplying (x-3)(x-3)

Answer 12

yes

Answer 13

(x-3)^2*(x^2-6x+10)

turns into

(x^2-6x+9)*(x^2-6x+10)

Answer 14

now expand out (x^2-6x+9)*(x^2-6x+10)

you can use the box method (which is what I would do)