Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
4, -14, and 5 + 8i

If I understand correctly I need to go
(x-4)(x+14)(x-(5+8i))(x+(5+8i))
but while I'm multiplying that out I must mess up because none of my answers match the multiple choice.

Answer 1

for (x-4)(x+14) I get x^2+10x-56 it's with the imaginaries that I start to falter.

Answer 2

Wait one of those would be x-(5-8i) right? I think I typed it wrong.

Answer 3

You wrote:
\((x-4)(x+14)(x-(5+8i))(x+(5+8i))\)

You are close, but the imaginary roots come in complex conjugate pairs, so you need to have root 5 + 8i and 5 - 8i.

You need to have (notice corrections in red):

\( \bf (x-4)(x+14)(x-(5+8i))(x\color{red}{-}(5\color{red}{-}8i))\)

Answer 4

That's what I had written down on my scratch paper, sorry. I copied it over wrong.
I'm still getting hung up on multiplication though... so many numbers :O

Answer 5

So I did that and now I have
(x^2+10x-56)(x^2-10x+89)
so far is that okay?

Answer 6

so now it says x^4-67x^2+1450x-4984 and that matches a multiple choice answer. So I think that's good, right?

Answer 7

\( (x-4)(x+14)(x-(5+8i))(x-(5 -8i))\)

\( = (x-4)(x+14)[x-5-8i][x-5 +8i]\)

\( = (x-4)(x+14)[(x-5)-8i][(x-5) +8i]\)

Notice that the last two factors (in square brackets) are now written as the product of a sum and a difference. \( (a + b)(a - b) = a^2 - b^2 \)

\( = (x-4)(x+14)[(x-5)^2-(8i)^2] \)

\( = (x-4)(x+14)[(x^2 - 10x + 25)-(-64) ] \)

\( = (x-4)(x+14)(x^2 - 10x + 89) \)

As you can see, there are no more imaginary numbers.

Answer 8

Yes, your answer is correct.

Answer 9

OK :) Thanks. Sometimes I make the dumbest errors and I just need help finding them to work them out.