Question

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

4, -2, and -1 + 2i

Answer 1

\[(x-4)(x+2)(x-(-1+2i))(x-(-1-2i))\]

Answer 2

the first part is routine
\[(x-4)(x+2)\] it is the second part that requires work, but i can show you a snap way to do it if you like

Answer 3

one way to find the quadratic with zero at \(-1+2i\) is to work backwards
put
\[x=-1+2i\] add 1 get
\[x+1=2i\] square and get
\[(x+1)^2=-4\] or
\[x^2+2x+1=-4\] or
\[x^2+2x+5\] that is your quadratic

Answer 4

the real snappy way is to memorize that if \(a+bi\) is the zero of a quadratic, the equation is
\[x^2-2ax+(a^2+b^2)\]

Answer 5

ok this was such a big help so you just multiplied like terms and then broke down the 2i andi just did it on paper and got x^2-2x-8 the first part of (x-4)(x+2)

Answer 6

i got -4x^2+8x+32 as the answer, correct?

Answer 7

i made an error, my final answer is f(x) = x4 - 7x2 - 26x - 40