Find a polynomial equation with real coecients that has −2 and i√2 as roots.

Question

Find a polynomial equation with real coecients that has −2 and i√2 as roots.

zepdrix · Answer

Hey April :)

If $\rm x=-2$ is a root of some polynomial,
adding 2 to each side of this equation gives us $\rm x+2=0$.
A polynomial which has -2 as a root will have $\rm (x+2)$ as a factor!

What we're doing is...
we're using the `roots` to form `factors`.
We'll then multiply these factors together to form our final `polynomial`.

zepdrix · Answer

Recall that complex roots always come in `pairs`,
conjugate pairs to be more precise.

So if $\rm x=i\sqrt2$ is a root,
then $\rm x=-i\sqrt2$ is also a root of the same polynomial.

zepdrix · Answer

We'll do the same addition or subtraction trick from before,
giving us this,$$\rm x-i \sqrt2=0,\qquad\qquad x+i \sqrt2=0$$So again, this gives us factors, $\rm (x-i\sqrt2)$ and $\rm (x+i\sqrt2)$,
which will allow us to build up our polynomial.

zepdrix · Answer

So to get your polynomial,
multiply these factors together,$$\large\rm (x+2)(x-i \sqrt2)(x+i \sqrt2)\quad=\quad ?$$