miley23 2 years ago Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 3, -13, and 5 + 4i A. f(x) = x4 - 8x3 - 12x2 + 400x - 1599 B. f(x) = x4 - 200x2 + 800x - 1599 C. f(x) = x4 - 98x2 + 800x - 1599 D. f(x) = x4 - 8x3 + 12x2 - 400x + 1599

1. ZeHanz

Remember, to have zeros 3, -13 and 5+4i, f has factors x-3, x+13 and x-5+14i. Also if 5+4i is a zero, then 5-4i is also a zero. So at least f should look like: f(x)=(x-3)(x+13)(x-5+14i)(x-5-14i). Now all you have to do is expand f to see which of A,B,C or D it is!

2. ZeHanz

3. miley23

D

4. ZeHanz

I got C

5. ZeHanz
6. satellite73

this part \((x-(5+4i))(x-(5-4i))\) can be a bit tricky. there are several ways to do it

7. satellite73

one way is to work backwards: \[x=5+4i\] \[x-5=4i\] \[(x-5)^2=-16\] \[x^2-10x+25=-16\] \[x^2-10x+41\] is the quadratic

8. satellite73

another way to to memorize that if the zeros are \(a+bi\) and \(a-bi\) then the quadratic with those zeros is \(x^2-2ax+a^2+b^2\)

9. satellite73

you can then go right to \((x-(5+4i))(x-(5-4i))=x^2-10x+5^2+4^2\)

10. satellite73

of course another way it to type it in to wolfram