Question

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

4, -2, and -1 + 2i
@ganeshie8

Answer 1

so confused

Answer 2

use this :
if \(k\) is a root, then \((x-k)\) is a factor of the polynomial

Answer 3

since \(4\) is a root, \((x-4)\) is a factor of the polynomial
...

Answer 4

can u write out the corresponding factors for other roots ?

Answer 5

(x-4)(x+2)(-1-2i)

Answer 6

good try, but what happened to x in the last factor ?

Answer 7

x+1-2i

Answer 8

since \(-1+2i\) is a root, \((x-(-1+2i))\) is a factor of the polynomial

Answer 9

ohhhh

Answer 10

actually it simplifies to x+1-2i only :) so you're right !

Answer 11

so the factored form of polynomial wid given roots is :

\((x-4)(x+2)(x-(-1+2i))\)

Answer 12

fine so far ?

Answer 13

yes

Answer 14

good, there is another little thing u need to rememeber :
complex roots always come in conjugate pairs

Answer 15

what that means is :
if \(a+bi\) is a one root, then its conjugate pair \(a-bi\) also will be a root

Answer 16

since you're given that \(-1+2i\) is one root, its conjugate pair \(-1-2i\) also will be a root

okay wud that ?

Answer 17

so far, yes im folowing

Answer 18

since \(-1-2i\) is a root, \((x - (-1-2i))\) is a factor

Answer 19

so i just factor the three?

Answer 20

So the full factored form of polynomial is :

\((x-4)(x+2)(x-(-1+2i))(x-(-2-2i)) \)

Answer 21

multiply them together

Answer 22

you get :
http://www.wolframalpha.com/input/?i=simplify+%28x-4%29%28x%2B2%29%28x-%28-1+%2B+2i%29%29%28x+-+%28-1-2i%29%29

Answer 23

the required polynomial is \(\large x^4 - 7x^2 - 26x - 40\)

Answer 24

you are literally the best on this site?

Answer 25

!*