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

Question

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

imstuck · Answer

This is a fourth degree polynomial because your roots are x-3, x+13, x-(5+4i), and x-(5+4i).  You multiply all those out to get your polynomial.

imstuck · Answer

$$(x-3)(x+13)(x-[5+4i])(x+[5+4i])$$

anonymous · Answer

i hope this helps http://www.mathportal.org/calculators.php

imstuck · Answer

oops made a type up above in my first reply.  One of those last roots is supposed to be x+[5+4i]

mathstudent55 · Answer

When a polynomial has real coefficients and complex roots, the complex roots come in complex conjugate pairs.
Since you are given the complex root 5 + 4i, then 5 - 4i must also be a root. That means the least number of roots is 4, and that means this is at least a 4th degree polynomial.
The roots are: 3, -13, 5 + 4i, and 5 - 4i 

The polynomial is:

$f(x)= (x - 3)[x - (-13)][x - (5 + 4i)][x - (5 - 4i)]$

Which simplifies to 

$f(x) = (x - 3)(x +13)(x - 5 - 4i)(x - 5 + 4i)$

anonymous · Answer

I see how you got that i just dont know how to distribute it because the final answer is supposed to be one of these:                                                      f(x) = x4 - 8x3 - 12x2 + 400x - 1599

f(x) = x4 - 200x2 + 800x - 1599

f(x) = x4 - 98x2 + 800x - 1599

f(x) = x4 - 8x3 + 12x2 - 400x + 1599
 and everytime i try i get something totally different

mathstudent55 · Answer

We'll do it together.

mathstudent55 · Answer

We need to multiply this out.

(x - 3)(x +13)(x - 5 - 4i)(x - 5 + 4i)

mathstudent55 · Answer

Let's start with the last two factors: (x - 5 - 4i)(x - 5 + 4i)
In order to make that multiplication a little easier, let's rewrite it a the product of a sum and a difference, so we get a difference of two squares.

mathstudent55 · Answer

$(x - 3)(x +13)[\color{red}{(x - 5)} - \color{blue}{4i}][\color{red}{(x - 5)} + \color{blue}{4i}]$

mathstudent55 · Answer

You see that the last two terms are the product of a sum and a difference?

anonymous · Answer

yea i see how u got that, the next step messes me up

mathstudent55 · Answer

Now let's just multiply together the last two terms using:
$ (a + b)(a - b) = a^2 - b^2$

mathstudent55 · Answer

$(x - 3)(x +13)[\color{red}{(x - 5)} - \color{blue}{4i}][\color{red}{(x - 5)} + \color{blue}{4i}]$

$= (x - 3)(x +13)[(x - 5)^2 - (4i)^2)] $

Ok?

mathstudent55 · Answer

$= (x - 3)(x +13)(x^2 - 10x + 25 - 16i^2)$

$= (x - 3)(x +13)(x^2 - 10x + 25 +16)$

$= (x - 3)(x +13)(x^2 - 10x + 41)$

Ok. We now multiplied the last two factors together.
Now let's multiply the first two factors together: (x - 3)(x +13)

$= (x^2 +13x -3x -39)(x^2 - 10x + 41)$

$= (x^2 +10x -39)(x^2 - 10x + 41)$

Ok so far?

mathstudent55 · Answer

The final step is to multiply those two trinomials together.
To do that, multiply every term, of the first one by every term of the second one. Then combine like terms.

mathstudent55 · Answer

$= (x^2 +10x -39)(x^2 - 10x + 41)$

$= x^4 - 10x^3 + 41x^2 + 10x^3 - 100x^2 + 410x - 39x^2 + 390x - 1599$

$f(x) = x^4 - 98x^2 + 800x - 1599$

anonymous · Answer

Thank you so much!! i honestly get it! thanx for helping step by step..you're awesome!

mathstudent55 · Answer

You're welcome.
I'm glad you understood it.