Question

Can anyone help me with this question, this is my last question I have to turn in.

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

5, -3, and -1 + 3i

I know that (x-5)(x+3), but I don't understand -1+3i. Any help will be great!

Answer 1

Here are the answers.

f(x) = x4 + 12.5x2 - 50x - 150

f(x) = x4 - 4x3 + 15x2 + 25x + 150

f(x) = x4 - 4x3 - 15x2 - 25x - 150

f(x) = x4 - 9x2 - 50x - 150

Answer 2

I also did (x-5)(x+3) = x^2 -2x - 15, aaaaaaand I'm stuck.

Answer 3

sorry cant help with this

Answer 4

5, -3, and -1 + 3i
usually means
(x-5) = 0 => x-5 =0 => x = 5
(x+3) = 0 => x+3 = 0 => x = -3
so
if we use say
x = -1 + 3i => x+1-3i =0 => (x+1-3i) =0

one thing to keep in mind is that, "complex roots" come in pairs in a polynomial
that that fellow isn't all by its lonesome, it has a conjugate companion

Answer 5

so, usually it'd be like
( (x+1) - 3i) is part of a "difference of a binomial"
like => \(\bf (a-b)(a+b) = (a^2-b^2)\)
so that one above is just the (a - b) part only
the companion will be the (a +b ) part
so
\(\bf ( (x+1) - 3i) ( (x+1) + 3i) = ( (x+1)^2 - (3i)^2) \)

Answer 6

So, would it be x^3-x^2-17x+3ix+45i-15(-x-1+3i)?

Answer 7

all 4 roots multiplied to each other, yes

Answer 8

in this case it'd be 2 real roots, and 2 complex ones, so the polynomial will be a 4 degree polynomial

Answer 9

I did the foil and I got x^4 - 9x^2 - 50x - 150

Answer 10

Wooohooo! Thanks!! Now I understand this concept!

Answer 11

so \(\bf ( (x+1)^2 - (3i)^2) \implies (x^2+2x+1)-(9)(i^2)\)
now \(i^2 = -1\)
so
\(\bf ( (x+1)^2 - (3i)^2) \implies (x^2+2x+1)-(9)(-1)\)

Answer 12

yw