Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
4, -14, and 5 + 8i

Answer 1

If 2 + 5 i is a root, them 2-5i is also.
(x−4)(x−(2+5i))(x−(2−5i))(x+8)

Do you know how to expand it?

Answer 2

ok the easy part is to start with
\[(x-4)(x+14)\] what will give you a polynomial whose zeros are \(4\) and \(-14\)
the harder part is coming up with a quadratic whose zeros are \(5+8i\) and \(5-8i\)
there are several ways to do it
i can show you an easy one, and a real real easy one if you like

Answer 3

can you not create a foil problem like you did with the first?

Answer 4

one way is to work backwards
put \[x=5+8i\] subtract \(5\) get
\[x-5=8i\] square both sides carefully and get
\[(x-5)^2=(8i)^2\] or \[x^2-10x+25=-64\] then add \(64\) to get
\[x^2-10x+89\] as your quadratic

Answer 5

there is an even snappier way, but it requires memorizing something
the quadratic with zeros \(a+bi\) and\(a-bi\) is
\[x^2-2ax+(a^2+b^2)\]
in your case \(5+8i\) you have \(a=5,b=8\) so it is
\[x^2-2\times 5x+(5^2+8^2)\] or
\[x^2-10x+89\] as before

Answer 6

final job is to multiply out
\[(x-4)(x+14)(x^2-10x+89\]
if it was me, i would cheat

Answer 7

hold on where did you just get
(or x2−10x+25=−64
then add 64 to get
x2−10x+89
as your quadratic)
this part from?

Answer 8

lets take it step by step
starting with \[x=5+8i\] which is what you were given

Answer 9

when you subtract 5 you get \[x-5=8i\] so far so good?

Answer 10

yes c:

Answer 11

now square both sides

Answer 12

x^2+25=64(-1)
=-64

Answer 13

on the right, you get \((8i)^2=8i\times 8i=8^2i^2=64(-1)=-64\)

Answer 14

but you have to square carefully on the left

Answer 15

\[(x-5)^2=(x-5)(x-5)=x^2-5x-5x+25=x^2-10x+25\]

Answer 16

oh okay you foil the left

Answer 17

i call it multiplying but i suppose you could say ...

Answer 18

in any case now you see why the left is
\[x^2-10x+25=-64\] and once you add \(64\) you get
\[x^2-10x+89\]

Answer 19

okay and then what

Answer 20

you gotta multiply out this ugly mess \[(x-4)(x+14)(x^2-10x+89)\]
like i said, i would cheat
want to check it?

Answer 21

one sec im working it our c:

Answer 22

k let me know when you want the answer

Answer 23

k so is it x^4-30x^3-487x^2+1450x-4984

Answer 24

http://www.wolframalpha.com/input/?i=%28x-4%29%28x%2B14%29%28x^2-10x%2B89%29

looks like it is \(x^4-67 x^2+1450 x-4984\)

Answer 25

hard to keep it all straight, that is why i said i would cheat
almost impossible not to make some kind of mistake

Answer 26

okay thanks

Answer 27

yw
use wolfram, it is your friend