(2n+1)(n^2-3n-1)-2n^3+1 Chia Hết Cho 5

2 min read Jun 16, 2024
(2n+1)(n^2-3n-1)-2n^3+1 Chia Hết Cho 5

Proving (2n+1)(n^2-3n-1)-2n^3+1 is Divisible by 5

This article will demonstrate that the expression (2n+1)(n^2-3n-1)-2n^3+1 is always divisible by 5, for any integer value of 'n'.

Expanding the Expression

First, let's expand the expression:

(2n+1)(n^2-3n-1)-2n^3+1 = 2n^3 - 6n^2 - 2n + n^2 - 3n - 1 - 2n^3 + 1

Combining like terms, we get:

= -5n^2 - 5n

Factoring out a 5

Now, we can factor out a 5 from the expression:

= 5(-n^2 - n)

Conclusion

As we can see, the expression is a product of 5 and another integer (-n^2 - n). Since it is a product of 5, it is always divisible by 5 for any integer value of 'n'.

Therefore, we have proven that (2n+1)(n^2-3n-1)-2n^3+1 is divisible by 5 for all integer values of 'n'.

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