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

you know it will look like \((x-4)(x+14)(some quadratic)\)

Answer 2

the hard part is finding the quadratic with zeros \(5+8i\) and \(5-8i\)
do you know how to do that?

Answer 3

i can show you two quick ways to do it if you like
one is pretty quick, the other is amazingly quick, although you have to memorize something

Answer 4

I'd love a demonstration, thanks!

Answer 5

ok one way is to work backwards
put \[x=4+8i\] subtract 4 get
\[x-4=8i\] square (carefully) and get
\[(x-4)^2=-64\] or
\[x^2-8x+16=-64\] then add \(64\) and get
\[x^2-8x+80=0\] and your quadratic is \(x^2-8x+80\)

Answer 6

ahhh damn damn damn, it was \(5+8i\) not \(4+8i\)

Answer 7

no matter, we can adjust quickly
\[x=5+8i\\
x-5=8i\\
(x-5)^2=-64\\
x^2-10x+25=-64\\
x^2-10x+89=0\]

Answer 8

really quick method requires memorizing that if \(a+bi\) is a zero of a quadratic, the quadratic is
\[x^2-2ax+(a^2+b^2)\]
in this case \(a=5, b=8\) so the quadratic is
\[x^2-2\times 5x+(5^2+8^2)=x^2-10x+89\]

Answer 9

your last job is to multiply
\[(x-4)(x+14)(x^2-10x+89)\] without making an algebra error
good luck with that, i would cheat

Answer 10

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

Answer 11

Wow, that was incredibly helpful. Thank you so much!

Answer 12

yw