Question

PLEASE HELP ME!!
find a third-degree polynomial equation with rational coefficients that has roots -2 and 2+i

would the root -2 become (x+2) or (x-2)?? i know i would multiply (x+(2+i))(x+(2-i)), because of conjugate pairs

Answer 1

Try \[x^3+6 x^2+13 x+10 \]

Answer 2

Hello Brittney!

Remember: COMPLEX ROOTS - roots like (2+i) - come in pairs! Where there's a (2+i), there's a (2-i). -2, as a real number, would be left alone. So what we're really looking for is a polynomial equation with the roots: -2, (2+i), (2-i). Cool!

So...

How do we get it?

Easy. You just plug these roots into factor form and multiply ALL the elements together in order to get the un-factored polynomial equation. In this case, we're multiplying:

(x+2) * (x -(2+i)) * (x-(2-i))

Let's start with the complex root section first.

(x -(2+i)) * (x-(2-i))

Okay? Let's distribute the negative signs.

(x - 2 - i) * (x - 2 +i)

Now do distribution of terms.

With x in the first pair of parentheses, we'll distribute it first:
x(x - 2 + i)
(x^2) - 2x + xi

Now distribute -2:
-2(x - 2 + i)
-2x +4 - 2i

Now, do it with -i:
-i(x - 2 + i)
-ix + 2i - (i^2)

Now, notice that when we distributed -i, we ended up with (i^2). Remember how (i^2) was equal to -1? Yeah! Now we can replace:

-ix + 2i - (i^2)

with:

-ix + 2i - (-1)

So it becomes:

-ix + 2i + 1.

Having distributed each of the terms in the first set of parentheses, we can add the distributed forms together!

We get:

( (x^2) - 2x + xi) + (-2x +4 - 2i) + (-ix + 2i + 1)

Let's solve!

This boils down to:

(x^2) - 4x + xi + 4 - 2i - ix + 2i + 1

Which is:

(x^2) - 4x + 4 - 2i + 2i + 1

Which is:

(x^2) - 4x + 4 + 1, or:

(x^2) - 4x + 5.

Okay, but remember the (x+2) that we left a long, long time ago in a land far, far away? We gotta multiply it back onto the simplified form of (x -(2+i)) * (x-(2-i)).

So:

(x+2)( (x^2) - 4x + 5)

is what we must multiply now.

Distributing, we get:

(x^3) + 2x^2 - 4x^2 - 8x + 5x + 10.

Simplify!

(x^3) - 2x^2 - 3x + 10.

This is what I think the answer is. Robtobey may be correct though, so definitely try to see how he got his answer!