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

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Answer 1 · 2017-10-05T17:41:38

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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