Can You Simplify This Complex Algebraic Expression?

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Discussion Overview

The discussion revolves around the simplification and factoring of the algebraic expression (x^2 + 1)^(3/2) + (x^2 + 1)^(7/2). Participants explore various approaches to factor and simplify the expression, focusing on the correct application of algebraic rules.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes factoring out (x^2 + 1)^(3/2) from the original expression.
  • Another participant elaborates on the factoring process, detailing how to subtract exponents correctly.
  • A subsequent post suggests further simplification of the expression 1 + (x^2 + 1)^2 and provides a step-by-step breakdown.
  • Another participant points out an oversight in the simplification, indicating that a term was omitted in the final expression.
  • One participant concludes with a different final answer, (x^2 + 1)^(3/2)(x^4 + 2x^2 + 2), which raises questions about the correctness of previous simplifications.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the final simplified form of the expression, and there are competing views on the correctness of the simplifications presented.

Contextual Notes

There are unresolved aspects regarding the accuracy of the final expressions and the steps taken to simplify them, as well as potential dependencies on interpretations of algebraic rules.

mathdad
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Factor

(x^2 + 1)^(3/2) + (x^2 + 1)^(7/2)

Solution:

(x^2 + 1)^(3/2)[1 + (x^2 + 1)^(7/12)]

Correct?
 
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When you factor out the expression $x^2+1$, you factor out the term having the smallest exponent (which you did), and then you subtract that exponent from the terms:

$$(x^2+1)^{\frac{3}{2}}+(x^2+1)^{\frac{7}{2}}=(x^2+1)^{\frac{3}{2}}\left((x^2+1)^{\frac{3}{2}-\frac{3}{2}}+(x^2+1)^{\frac{7}{2}-\frac{3}{2}}\right)=(x^2+1)^{\frac{3}{2}}\left((x^2+1)^{0}+(x^2+1)^{\frac{4}{2}}\right)=(x^2+1)^{\frac{3}{2}}\left(1+(x^2+1)^{2}\right)$$
 
(x^2 + 1)^(3/2)(1 + (x^2 + 1)^2)

What about simplifying the right expression more?

Right Expression:

1 + (x^2 + 1)^2

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

1 + x^4 + 2x^2 + 1

x^4 + 2x^2 + 1

Final answer:

(x^2 + 1)^(3/2)(x^4 + 2x^2 + 1)

Correct?
 
RTCNTC said:
(x^2 + 1)^(3/2)(1 + (x^2 + 1)^2)

What about simplifying the right expression more?

Right Expression:

1 + (x^2 + 1)^2

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

1 + x^4 + 2x^2 + 1

x^4 + 2x^2 + 1

You've dropped one of the 1's there...:D
 
Answer: (x^2 + 1)^(3/2)(x^4 + 2x^2 + 2)
 

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