we were given a function of y=(x-a)(x-b)(x-c) and were told to add in our own numbers i chose y=(x-8)(x+5)(x-3). can some one show me the working to get it to a cubic and then the derivative
lets say it was (x - 8)(x + 5), could you expand that?
expand the first two first, and then multiply what you got by the (x - 3). how? first multiply everything by the x and then multiply everything by the -3
when finding the derivative, all that is needed is the power rule: $\displaystyle \frac d{dx} x^n = nx^{n - 1}$
and of course, if there are constants in front of the x's, you just leave them and multiply what you got by them since $\displaystyle \frac d{dx}[c f(x)] = c \frac d{dx}f(x)$. here, your $\displaystyle f(x)$ is $\displaystyle x^n$
(why did you choose such big numbers? )
$\displaystyle \underbrace{(x - a)(x - b)}_{\text{do these two first}}(x - c) = [x^2 - (a + b)x + ab](x - c)$
now we take the x in the second set of brackets and multiply everything in the first set, after that, we take the -c in the second set of brackets and multiply everything in the first set. so we get:
$\displaystyle \underbrace{x^3 - (a + b)x^2 + abx}_{\text{everything in the first set times }x}~~~ \underbrace{- cx^2 + c(a + b)x - abc}_{\text{everything in the first set times }-c} $
now group the like terms and simplify
$\displaystyle x^3 - (a + b + c)x^2 + (ab + ac + bc)x - abc$