Can you please explain how to factorize

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Discussion Overview

The discussion revolves around the factorization of the polynomial x4 + 2x2 + 9. Participants explore various methods of factorization, including the quadratic formula and alternative algebraic manipulations, while addressing the nature of the roots involved.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants note that using the quadratic formula yields complex roots for the polynomial x4 + 2x2 + 9.
  • Others reference a book that provides a factorization of the polynomial as (x2 + 2x + 3)(x2 - 2x + 3).
  • A participant emphasizes that factorization over the reals requires factors to have real coefficients, highlighting the relationship between complex roots and their conjugates.
  • One participant presents an alternative algebraic manipulation to express the polynomial as (x2 + 3 + 2x)(x2 + 3 - 2x), suggesting a different approach to factorization.
  • Another participant proposes substituting u = x2 to simplify the problem to solving u2 + 2u + 9.

Areas of Agreement / Disagreement

Participants express differing views on the factorization methods and the nature of the roots, indicating that multiple competing approaches and interpretations exist without a clear consensus.

Contextual Notes

Some participants mention the requirement for real coefficients in factorization, and the discussion includes various algebraic techniques that may not be universally accepted or resolved.

sarah786
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Can you please explain how to factorize x4+2x2+9
If i do it by quadratic formula, i get complex roots ... in my book, it has been factorized to
(x2+2x+3)(x2-2x + 3)
 
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sarah786 said:
Can you please explain how to factorize x4+2x2+9
If i do it by quadratic formula, i get complex roots ... in my book, it has been factorized to
(x2+2x+3)(x2-2x + 3)

Which complex roots did you get?
 


sarah786 said:
Can you please explain how to factorize x4+2x2+9
If i do it by quadratic formula, i get complex roots ... in my book, it has been factorized to
(x2+2x+3)(x2-2x + 3)

When it says factorize over the reals, it means your factors have to have real coefficients. A root of x-(a+bi) does not have all real coefficients. Also remember that any real quadratic with a complex root also has a complex conjugate as its other root. So you can also work backwards from the two conjugates to find the real quadratic that has those roots.
 


your equation can be written as
X4 + 6X2 - 4X2 + 9
= X4 + 6X2+9 - 4X2
= ( X2 + 3 )2 - (2X)2
= ( X2+3+2X) (X2+3- 2X)
 
Last edited:


sarah786 said:
Can you please explain how to factorize x4+2x2+9
If i do it by quadratic formula, i get complex roots ... in my book, it has been factorized to
(x2+2x+3)(x2-2x + 3)
Let u=x2. Then you need to solve u2 +2u +9, and take square roots of both solutions.
 

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