Question

Simplify `30+114n -114 +81{(n-1)(n-2)} + 17{(n-1)(n-2)(n-3)} + (n-1)(n-2)(n-3)(n-4)`?

Answer 1

`n(n+1)(n+2)(n+4)` or `n^4+7n^3+14n^2+8n`

Explanation:

`30+114*(n-1)+81(n-1)(n-2)+17(n-1)(n-2)(n-3)+(n-1)(n-2)(n-3)(n-4)`

After using `y=n-1` transform, this polynomial became

`30+114y+81y(y-1)+17y(y-1)(y-2)+y(y-1)(y-2)(y-3)`

=`30+114y+81y^2-81y+17*(y^3-3y^2+2y)+(y^2-y)*(y^2-5y+6)`

=`30+81y^2+33y+17y^3-51y^2+34y+y^4-6y^3+11y^2-6y`

=`y^4+11y^3+41y^2+61y+30`

=`(n-1)^4+11(n-1)^3+41(n-1)^2+61*(n-1)+30`

=`n^4-4n^3+6n^2-4n+1+11n^3-33n^2+33n-11+41n^2-82n+41+61n-61+30`

=`n^4+7n^3+14n^2+8n`

=`n*(n^3+7n^2+14n+8)`

=`n*(n^2+n+6n^2+6n+8n+8)`

=`n*(n+1)*(n^2+6n+8)`

=`n(n+1)(n+2)(n+4)`