Factoring question! Need help!?

The answer key is correct

Let f(x) = expression

For the expression to be divisible by x+4, f(-4) should be zero

So:

f(-4) = (-4)^5 + K(-4)^4 +25(-4)^3 - 20(-4)^2 - 80(-4) + 64 = 0

- 1024 + 256K - 1600 - 320 +320+ 64 = 0

256K = 1024 + 1600 +320 - 320 - 64 =2560

K = 2560/256 = 10

shrewwed and jerome (spelling) are both correct, but here's another point of view that you could try next time (if you have a spreadsheet)

In excel,

In A1 put: x

In B1 put: y

In A2 put: -4

In B2 put: =x^5 + 10 *x^4 +25*x^3 - 20*x^2 - 80*x +64

You'll see that B2 is 0 (zero) so the + 10 is correct.

That is:

If (x + 4) is a factor, then when x = -4 the polynomial must = 0

Spreadsheets are handy for lots of questions like this. (If you ever want to use 'Guess & Check' you just can't beat them.)

Set it up with synthetic division

-4 |...1...k...25...-20...-80...64

....|____-4_-4k+16_____________

......1.(k-4)

I could show it better on paper, but i hope you get the idea.

At the very end of the equation, you want to ensure no remainder.

Whatever value of k makes the last term 0, that is the answer.