Question

How do you simplify `(t^5 - 3t^2 - 20) (t-2)^-1`?

Answer 1

`(t^5-3t^2-20)(t-2)^(-1)=t^4+2t^3+4t^2+5t+10`

Explanation:

We can write `(t^5-3t^2-20)(t-2)^(-1)` as

`(t^5-3t^2-20)/(t-2)`

Now observe that for the numerator `f(t)=t^5-3t^2-20`,

`f(2)=2^5-3*2^2-20=32-12-20=0` and from factor theorem `(t-2)` is a factor of `t^5-3t^2-20`

and hence we can divide `t^5-3t^2-20` by `t-2` and

`t^5-3t^2-20`

= `t^4(t-2)+2t^3(t-2)+4t^2(t-2)+5t(t-2)+10(t-2)`

= `(t-2)(t^4+2t^3+4t^2+5t+10`

and hence `(t^5-3t^2-20)(t-2)^(-1)=t^4+2t^3+4t^2+5t+10`