Question

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

4, -8, and 2 + 3i

Answer 1

\[(x-4)(x+8)(x-(2+3i))(x-(2-3i))\] is a start

Answer 2

first product is easy enough, you get
\[(x-4)(x+8)=x^2-4x-32\] it is the second that looks difficult, but it isn't

Answer 3

for the other 2 how do you work that out?

Answer 4

\[(x-(2+3i))(x-(2-3i))\] use
first \(x^2\)
last \(2+3i)(2-3i)=2^2+3^2=4+9=13\)
outer and inner you will get \((-2+3i)x+(-2-3i)x\) and you see you are left with only \(-4x\)

Answer 5

don't forget that when you multiply a complex number \(a+bi\) by its complex conjugate \(a-bi\) you get \(a^2+b^2\) a real number, so last term is easy, you can do it in your head

Answer 6

then you can see that \(-(a+bi)x+-(a-bi)x=-2ax\)

Answer 7

woah you lost me

Answer 8

so last two terms give
\[x^2-4x+13\]

Answer 9

oh ok the way i was reading it was confusing

Answer 10

you know you have to do this multiplication
\((x-(2+3i))(x-(2-3i))\) right?

Answer 11

yea the product of that gives you -4x

Answer 12

i hate to say "foil" but that is the idea, first outer inner last
first is \(x^2\)
last is \(2^2+3^2\) and outer plus inner is \(-4x\)

Answer 13

finally you get to multiply
\[(x^2-4x-32)(x^2-4x+14)\] to finish

Answer 14

yea i undertand i was just reading it weird

Answer 15

ok have fun

Answer 16

then you multiply (x2 - 4x +32)(x2-4x+13) right?

Answer 17

how do you multiply them?

Answer 18

@satellite73

Answer 19

Good ol' distributive property, just go term by term and group like terms together after multiplying.

Answer 20

It's no different than multiplying 123 by 456 by expressing the product as
(100 + 20 + 3) × (400 + 50 + 6)
=100×400 + 100×50 + 100×6 + 20×400 + 20×50, etc.