Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write in standard form. 4, -8, and 2 + 5i
I got up to (x-4)(x+8)(2+5i)(2-5i) What do I do next?

Answer 1

there is a mistake there

Answer 2

the last to factors should be
\[(x-(2+5i))(x-(2+5i))\] there is an easy way to find this however, so it is not as bad as it looks

Answer 3

*last TWO factors

Answer 4

\[(x-4)(x+8)\] is routine, what you need is the polynomial of degree two with the two zeros \(2+5i\) and \(2-5i\) right?

Answer 5

i can show you how to do this more or less in your head if you like

Answer 6

Oh please do, but do I multiply them all together?

Answer 7

oh no i wouldn't do that

Answer 8

first of all we can work backwards
that is one way
put
\[x=2+5i\] subtract 2 and get
\[x-2=5i\] square and get
\[x^2-4x+4=-25\] add 25 and get
\[x^2-4x+29\] as your needed polynomial

Answer 9

an even easier way to do it, is to memorize a simple fact: if \(a+bi\) is a zero of a quadratic, then it is \(x^2-2ax+a^2+b^2\)

Answer 10

so if \(2+5i\) is a zero, the quadratic is \(x^2-2\times 2x+2^2+5^2=x^2-4x+29\)

Answer 11

your math teacher might want you to multiply all that out, and it is not that hard if you do it correctly, group terms etc
you will get the same answer of course, but who needs all that aggravation?

Answer 12

you are not finished the problem, you still have to multiply
\[(x-4)(x+8)(x^2-4x+29)\]

Answer 13

Is there any easy way to multiply that all out?

Answer 14

yes do this
http://www.wolframalpha.com/input/?i=expand+%28x-4%29%28x%2B8%29%28x^2-4x%2B29%29

Answer 15

you get
\[x^4-19 x^2+244 x-928\] with the click of a button
other than that, no
just have to grind it til you find it

Answer 16

Thank you soo much :)

Answer 17

yw