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 + 2i

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 + 2i

anonymous · Answer

helpp

anonymous · Answer

If $-1+2i$ is a root, then so is $-1-2i$.

anonymous · Answer

yeah! i dont know how to multiply (x-(-1+2i))(x-(-1-2i))

anonymous · Answer

Hold on give me a second...

anonymous · Answer

Look at the quadratic formula $$
x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}
$$Supposing that $a=1$ then clearly $b=2$ and $b^2-4ac = -16$

anonymous · Answer

I'm trying to reverse engineer the answer here.

anonymous · Answer

$$
4-4c = -16 \implies 1-c=-4\implies c=-5
$$

anonymous · Answer

Check to see if $x^2+2x-5$ has the roots we want.

anonymous · Answer

Woops,  $c=5$
And checking $x^2+2x+5$ gives us the correct roots of $-1-2i,-1+2i$

anonymous · Answer

yeahh

anonymous · Answer

$$
\begin{split}
(x-(-1+2i))(x-(-1-2i)) 
&= x^2 - (-1-2i)x - (-1+2i)x +(-1+2i) (-1-2i)\
&= x^2 +x+2ix +x-2ix + 1-4i^2 \
&= x^2+2x + 1-4(-1) \
&= x^2+2x+5

\end{split}
$$Multiplying them works well too.

anonymous · Answer

So in general, when you have an imaginary number, say it is $a+bi$
Then it should turn out to be: $$
x^2-2ax+a^2+b^2
$$

anonymous · Answer

hmmm 
 f(x) = x4 - 4x3 + 10x2 + 20x + 75

 f(x) = x4 - 14x2 - 40x - 75

 f(x) = x4 - 4x3 - 10x2 - 20x - 75

 f(x) = x4 + 10x2 - 40x - 75
these are my answer choice though...@wio

anonymous · Answer

@wio

anonymous · Answer

$$
(x-5)(x+3)(x^2+2x+5)
$$

anonymous · Answer

oohhh okay!!