Question

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

Answer 1

start with
\[(x-5)(x+3)(x-(-1+3i))(x-(-1-3i))\] and multiply out
that is the hard way to do it, but there is a somewhat easier way

Answer 2

whats easier?

Answer 3

you have no choice but to multiply
the first product is not hard
\[(x-5)(x+3)=x^2-2x-15\]
it is the second product that has an easier way to find then straight out multiplication

Answer 4

yeah that's the part i dont understand

Answer 5

you know one of the zeros is \(-1+3i\) so you can find the quadratic polynomial using one of two simple methods

Answer 6

start with
\[x=-1+3i\] and work backwards

Answer 7

\[x=-1+3i\]
\[x+1=3i\]
\[(x+1)^3=(3i)^2=-9\]
\[x^2+2x+1=-9\]
\[x^2+2x+10=0\]

Answer 8

that is the polynomial with zeros \(-1+3i\) and \(-1+3i\)

Answer 9

alternatively, you can simply remember that if \(a+bi\) is the zero of a quadratic polynomial with leading coefficient 1, that polynomial is
\[x^2-2a+(a^2+b^2)\]

Answer 10

so then just factor the two polynomials?

Answer 11

in your case \(a=-1,b=3\) so the polynomial must be
\[x^2+2x+10\]

Answer 12

no no don't "factor" you have it in factored form to begin with
your job is to multiply out, not factor

Answer 13

\[(x^2-2x-15)(x^2+2x+10)\] is what you have to compute to write the polynomial in standard form

Answer 14

thats what i meant! lol sorry