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

here are my choices
f(x) = x^4 - 7x^2 - 26x - 40
f(x) = x^4 - 3x^3 - 8x^2 - 13x - 40
f(x) = x^4 + 6.5x^2 - 26x - 40
f(x) = x^4 - 3x^3 + 8x^2 + 13x + 40

Answer 1

this again
ready?

Answer 2

i dont know how to do this at all lolol

Answer 3

ok then we better go slow

Answer 4

do you know if \(r\) is a zero, then one factor must be \((x-r)\)? so in this case you have \(4\) is a zero making one factor \((x-4)\)

Answer 5

clear or not? that is really what we need to start this

Answer 6

oh okay so (x-4) and (x+2) ?

Answer 7

right

Answer 8

we start with \[(x-4)(x+2)\] now we have more work to do

Answer 9

you also need the quadratic that has the zeros of \(-1+2i\) and it "conjugate" \(-1-2i\)
next job is to find that

Answer 10

any ideas? "no" is a fine answer

Answer 11

no i dont know

Answer 12

ok there is a hard way, an easy way, and a really really easy way
lets do the easy way first, then the real easy way

Answer 13

work backwards starting with \[x=-1+2i\] add \(1\) to get \[x+1=2i\]

Answer 14

then square both sides (carefully) to get \[(x+1)^2=(2i)^2\] or \[x^2+2x+1=-4\]

Answer 15

add 4 to both sides to finish with \[x^2+2x+5\]

Answer 16

final job is to multiply \[(x-4)(x+2)(x^2+2x+5)\]i would cheat so as not to screw up the algebra

Answer 17

http://www.wolframalpha.com/input/?i=%28x-4%29%28x%2B2%29%28x^2%2B2x%2B5%29

Answer 18

and would that be the answer?
or is there more

Answer 19

OOO got it. thank you

Answer 20

yeah that is it

Answer 21

yw