Question

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

3, -13, and 5 + 4i

Answer 1

Answers:
f(x) = x4 - 8x3 - 12x2 + 400x - 1599
f(x) = x4 - 200x2 + 800x - 1599
f(x) = x4 - 98x2 + 800x - 1599
f(x) = x4 - 8x3 + 12x2 - 400x + 1599

Answer 2

@welshfella

Answer 3

This one is really confusing me

Answer 4

Ok Do you know that all complex roots occur as conjugate pairs?

Answer 5

Yes

Answer 6

So the remaining root is 5 - 4i

so we have

f(x) = (x - 3)(x + 13)( x - (5+4i)) ( x - ( 5 - 4i))

Answer 7

Alright

Answer 8

expanding that is a bit of a pain , I know!

Answer 9

Yeah definitely haha

Answer 10

So then, I'm confused at what to do next

Answer 11

work out the 2 complex factors first

Answer 12

I got (x-3)(x+13)(x-5-4i)(x-5+4i)
I think I did something wrong, that doesn't seem right..

Answer 13

I actually got 41 by working out the 2 roots

Answer 14

you should be able to eliminate the imaginary parts

Answer 15

yes so that makes it (x^2 + 41)

Answer 16

Alright

Answer 17

- see what you mean though - the expansion doesnt look right though - Maybe I got it wrong....

Answer 18

This question was very confusing to me and didn't make much sense

Answer 19

yea I am a bit confused myself lol

I cant see anything wrong with my algebra though...

Answer 20

So what would be my best bet with this question, it is my last one haha

Answer 21

well expand
(x - 3)(x + 13)(x^2 + 41) and hopefully it wil give 1 of choices

Answer 22

wait its not x^2 + 41 we must have a term in x in it!!

Answer 23

lets do it on a scribble pad

Answer 24

yea it expands to

x^2 - 5x + 4xi - 5x + 25 - 20i -4xi + 20i - 16 i^2

Answer 25

there you go
You'll see that you lose all the i's when you simplify that.

Answer 26

remember that i^2 = -1

Answer 27

so - 16i^2 = -16 * -1 = + 16

Answer 28

so can you simplify it?

Answer 29

Let me try real quick

Answer 30

Not sure if this is right but I got -224

Answer 31

Yeah that doesn't seem right

Answer 32

???

you should end up with a trinomial something like x^2 - ax + 41

Answer 33

I'm not sure what I did..

Answer 34

the terms ending in i cancel out and last term = 25 + 16 = 41
and youl'l have a term in x

Answer 35

Just add like terms

Answer 36

Okay

Answer 37

x^2 - 5x + 4xi - 5x + 25 - 20i -4xi + 20i - 16 i^2
= x^2 - 5x - 5x + 25 + 16

Answer 38

4xi and 20i disappear

Answer 39

Okay, I get that now

Answer 40

so now you have to expand
(x - 3)(x + 13)(x^2 - 10x + 41)

Answer 41

yes, I did that up there and got the same trinomial

Answer 42

note that last term will be -3 + 13 * 41 = -1599 so you can eliminate D

Answer 43

I just got that too

Answer 44

x^4-98x^2+800x-1599

Answer 45

C?

Answer 46

= ( x^2 + 10x - 39)(x^2 - 10x + 41)

Answer 47

yes thats it
notice the terms in x^3 are + 10x^3 and - 10x^3 so that eliminates them

Answer 48

Thanks! It was right

Answer 49

yw

Answer 50

these are very tedious to work out and its easy to make a mistake.

Answer 51

I was too lazy to do the expansion!