(n^3+4n^2+3n)(n^3+5n^2)

Jasonbradley

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Jun 16, 2024

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Simplifying and Factoring (n^3+4n^2+3n)(n^3+5n^2)

This expression represents the product of two polynomial expressions. To understand it better, we can simplify and factor it.

Simplifying the Expression

1. Factor out common terms:

   • In the first polynomial, we can factor out 'n': n(n^2 + 4n + 3)
   • In the second polynomial, we can factor out 'n^2': n^2(n + 5)

2. Rewrite the expression: The original expression now becomes: n(n^2 + 4n + 3) * n^2(n + 5)

3. Multiply the factored terms:

   • n * n^2 = n^3
   • (n^2 + 4n + 3) * (n + 5) = n^3 + 5n^2 + 4n^2 + 20n + 3n + 15 = n^3 + 9n^2 + 23n + 15

Therefore, the simplified expression is: n^3(n^3 + 9n^2 + 23n + 15)

Factoring the Expression

The simplified expression can be factored further.

1. Factor the trinomial: The trinomial (n^3 + 9n^2 + 23n + 15) can be factored by grouping:

   • n^3 + 9n^2 + 23n + 15 = (n^3 + 9n^2) + (23n + 15)
   • = n^2(n + 9) + (23n + 15)

   Unfortunately, this trinomial cannot be factored further using simple methods.

2. Final factored form: The fully factored expression is: n^3(n^2(n + 9) + (23n + 15))

Conclusion

The expression (n^3+4n^2+3n)(n^3+5n^2) can be simplified and factored. While the trinomial (n^3 + 9n^2 + 23n + 15) cannot be factored further using simple methods, the final factored form is n^3(n^2(n + 9) + (23n + 15)).