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

f(x) = x4 - 362.5x2 + 1450x - 4984

f(x) = x4 - 9x3 + 32x2 - 725x + 4984

f(x) = x4 - 67x2 + 1450x - 4984

f(x) = x4 - 9x3 - 32x2 + 725x - 4984

Answer 1

\[x-4)(x+14)(quadratic)\] where 'quadratic' has zeros of \(5+8i\) and \(5-8i\) which is the only hard part, and it is not that hard to find

Answer 2

you have three choices. one is hard, one is easy and one is really really easy
you pick

Answer 3

I'm really confused still, what?

Answer 4

@satellite73

Answer 5

you are given three zeros right?

Answer 6

4, -14, and 5 + 8i

Answer 7

in fact you are actually given 4 zeros, because if \(5+8i\) is a zero, then so is its conjugate \(5-8i\)

Answer 8

since 4 is a zero, one factor of the polynomial is \(x-4\)

Answer 9

since \(-14\) is a zero, one factor of the polynomial is \(x-(-14) =x+14\)

Answer 10

clear so far or no?

Answer 11

@satellite73 yeah i get that!

Answer 12

then the next part is to find a quadratic with zeros of \(5+8i\) and \(5-8i\)

Answer 13

@satellite73 x^2-10x-80i+89???

Answer 14

whoa hold the phone
there should be no \(i\) in your answer

Answer 15

you have a choice
one way is to work backwards
\[x=5+8i\\
x-5=8i\] then square (carefully) and get
\[(x-5)^2=(8i)^2\\
x^2-10x+25=-16\]

Answer 16

and so your quadratic is
\[x^2-10x+41\]

Answer 17

oh ok @satellite73

Answer 18

then multiply all that mess out
\[(x+4)(x-14)(x^2-10x+41)\]

Answer 19

@satellite73 I got x^4-10x^3-25x^2+100x-2146?

Answer 20

i would use this so because otherwise i am sure i would make an algebra mistake
http://www.wolframalpha.com/input/?i=%28x%2B4%29%28x-14%29%28x^2-10x%2B41%29

look under "expanded form"

Answer 21

oh crap i made a mistake!!

Answer 22

guess i don't know how to square myself
\[(x-5)^2=(8i)^2\\
x^2-10x+25=-16\] is wrong
\[(x-5)^2=(8i)^2\\
x^2-10x+25=64\] is more like it

Answer 23

yeah i was wondering because that's not an answer choice

Answer 24

so
\[x^2-10x+89\] is your polynomial

Answer 25

for which one? @satellite73

Answer 26

the one with the two complex zeros
now go to \[(x+4)(x-14)(x^2-10x+89)\]

Answer 27

there we go
http://www.wolframalpha.com/input/?i=%28x%2B4%29%28x-14%29%28x^2-10x%2B89%29

Answer 28

btw if you ever have to do another one of these, and are told that a zero is \(a+bi\) then the quadratic is
\[x^2-2ax+(a^2+b^2)\]

Answer 29

I'm confused because that's still none of my answer choices @satellite73

Answer 30

we had \(5+8i\) and the quadratic was
\[x^2-2\times 5x+(5^2+8^2)\\
x^2-10x+25+64\\
x^2-10x+89\]

Answer 31

yeah because i am really an idiot today
i wrote
"if 4 is a zero then \(x-4\) is a factor, but then wrote the factor as \(x+4\)which is wrong
lets try one more time\[(x-4)(x+14)(x^2-10x+89)\]

Answer 32

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

Answer 33

That's more like it! Thank you soooo much for taking the time to help me!(:

Answer 34

i sure as hell hope \[x^4-20 x^3+133 x^2-330 x-4984\]is an answer choice!

Answer 35

yw
sorry i screwed up twice, but i hope you get the idea