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Answer 1 · 2017-06-13T21:19:34

$x^4-10x^3+20x^2+40x-96$

If we already have the zeros they original came from solving for x:
This is what we have right now $x=-2,2,4,6$
So we work our way backwards:

$x=-2$ You add two to both sides
$x+2=cancel(-2+2)$
$color(blue)(x+2)$ This is one factor

$x=2$ Subtract two from both sides
$x-2=cancel(2-2)$
$color(blue)(x-2)$ Another factor

$x=4$ Subtract four from both sides
$x-4=cancel(4-4)$
$color(blue)(x-4)$ Third factor

$x=6$ Subtract six from both sides
$x-6=cancel(6-6)$
$color(blue)(x-6)$ Final factor

Now that we know this information we can now multiply out our factors of:

$(x+2)(x-2)(x-4)(x-6)$

$(x^2-4)(x-4)(x-6)$

$(x^3-4x^2-4x+16)(x-6)$

$x^4-6x^3-4x^3+24x^2-4x^2+24x+16x-96$

Simplify:

$x^4-10x^3+20x^2+40x-96$