Question

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

8, -14, and 3 + 9i

Answer 1

\((x-8)(x+14)(x-(3+9i))(x-(3-9i))\) is a start
it is not as bad as it looks

Answer 2

It will be easier if you multiply the two product terms containing complex numbers first.

also, the reason that @satellite73 has you multiplying 4 terms even though only 3 zeros were specified is that any polynomial with only real numbers as coefficients must have complex zeros, if any, in conjugate pairs (\(a\pm bi, i = \sqrt{-1}\)).

Answer 3

oh i see, nvm

Answer 4

in any case, it is easy to find the last product

Answer 5

you don't actually have to multiply out, you can work backwards
\[x=3+9i\]
\[x-3=9i\]
\[(x-3)^2=-81\]
\[x^2-6x+3=-81\]
\[x^2+6x+84=0\]

Answer 6

let me rephrase: it will be easiest if you combine the last two terms first, whether by straightforward multiplication or actually thinking a little bit, like @satellite73 does :-)

Answer 7

actually the easiest method is simply to remember if \(a+bi\) is a root of quadratic with leading coefficient 1, that quadratic must be
\[x^2-2ax+(a^2+b^2)\]

Answer 8

of course that requires "memorizing" so maybe other methods are better

Answer 9

sometimes when we say "remember" we forget that that implies once having known :-)

Answer 10

good point

Answer 11

maybe i should rephrase
if you are going to have to do this with some speed on say a quiz or exam, might be best two write it on your shoe so you can cheat quickly, then wash the shoes