cubic functions? root? (brothers homework)

[sports_brah]

I feel bad asking, but I feel even worse for my young brother who is sweating like a sinner in church in regards tomorrow.

Does anyone have any tips/solutions/guides? Thank you!

 

[Scotty123]

If -k is a root, then that expression should be zero if you set x = -k.

(-k)^3 + (1 - k^2)(-k) + k = (-k)^3 - k + k^3 + k = 0.

As for the second part, probably polynomial division. Divide f by x + k to get a quadratic which will easily factorise. I'm not typing that out on an iPad!

 

[AGD]

The expression is:

f(x) = x^3 + (1-k^2)x + k

And we want to show that if x = -k then f(x) = 0, so inserting -k everywhere there is an x gives:

f(-k) = -k^3 + (1 - k^2) * -k + k
= -k^3 - k + k^3 + k
= 0

 

[sports_brah]

Thanks guys, he really appreciates it! tut tut, when I was 16, I wasn't like this.......