(x+1)^4+(x^2+x+1)^2

Jasonbradley

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Jun 16, 2024

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Expanding and Simplifying (x+1)^4 + (x^2 + x + 1)^2

This article will explore the expansion and simplification of the algebraic expression (x+1)^4 + (x^2 + x + 1)^2.

Expanding the Expressions

We can start by expanding each term individually.

• (x+1)^4: This can be expanded using the binomial theorem or by repeated multiplication:

  • Binomial Theorem: (x+1)^4 = 1x^4 + 4x^31 + 6x^21^2 + 4x1^3 + 11^4 = x^4 + 4x^3 + 6x^2 + 4x + 1
  • Repeated Multiplication: (x+1)^4 = (x+1)(x+1)(x+1)(x+1) = (x^2 + 2x + 1)(x^2 + 2x + 1) = x^4 + 4x^3 + 6x^2 + 4x + 1

• (x^2 + x + 1)^2: This can be expanded by squaring the binomial:

  • (x^2 + x + 1)^2 = (x^2 + x + 1)(x^2 + x + 1) = x^4 + 2x^3 + 3x^2 + 2x + 1

Combining the Expanded Terms

Now that we have expanded both terms, we can combine them:

(x+1)^4 + (x^2 + x + 1)^2 = (x^4 + 4x^3 + 6x^2 + 4x + 1) + (x^4 + 2x^3 + 3x^2 + 2x + 1)

Simplifying the Expression

Finally, we can simplify the expression by combining like terms:

(x^4 + 4x^3 + 6x^2 + 4x + 1) + (x^4 + 2x^3 + 3x^2 + 2x + 1) = 2x^4 + 6x^3 + 9x^2 + 6x + 2

Conclusion

Therefore, the simplified form of (x+1)^4 + (x^2 + x + 1)^2 is 2x^4 + 6x^3 + 9x^2 + 6x + 2.