MHB How Do You Factor \(x^8+4x^2+4\) into Non-Constant Polynomials?

  • Thread starter Thread starter anemone
  • Start date Start date
AI Thread Summary
The polynomial \(x^8+4x^2+4\) can be factored into two non-constant polynomials with integer coefficients. Members Olinguito, Opalg, and kaliprasad provided correct solutions to the problem. Olinguito presented a primary solution, while Opalg offered an alternate approach. The discussion emphasizes the importance of following the guidelines for problem-solving in the forum. Engaging with such problems enhances understanding of polynomial factorization techniques.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Here is this week's POTW:

-----

Factor $x^8+4x^2+4$ into two non-constant polynomials with integer coefficients.

-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
Congratulations to the following members for their correct solution!(Cool)

1. Olinguito
2. Opalg
3. kaliprasad

Solution from Olinguito:
$$x^8+4x^2+4$$
$=\ (x^8+4x^6+6x^4+4x^2+1)-4x^6-6x^4+3$

$=\ (x^2+1)^4-4x^6-6x^4+3$

$=\ [(x^2+1)^4+2(x^2+1)^2+1]-4x^6-8x^4-4x^2$

$=\ [(x^2+1)^2+1]^2-4x^2(x^4+2x^2+1)$

$=\ [(x^2+1)^2+1]^2-4x^2(x^2+1)^2$

$=\ ([(x^2+1)^2+1]-2x(x^2+1))([(x^2+1)^2+1]+2x(x^2+1))$

$=\ (x^4-2x^3+2x^2-2x+2)(x^4+2x^3+2x^2+2x+2)$.


Alternate solution from Opalg:
If $x$ is a solution of the equation $x^8 + 4x^2 + 4 = 0$ then so is $-x$. So the eight (complex) roots of that equation can be split into two groups of four, with each root in the second group being the negative of one of the roots in the first group. If $x^4 + ax^3 + bx^2 + cx + d = 0$ is the equation whose roots are the first group of four, then (replacing $x$ by $-x$) $x^4 - ax^3 + bx^2 - cx + d = 0$ is the equation whose roots are the second group. Therefore $$(x^4 + ax^3 + bx^2 + cx + d)(x^4 - ax^3 + bx^2 - cx + d) = x^8 + 4x^2 + 4.$$ Now compare the coefficients of powers of $x$ on both sides of that equation: $$ x^6: \quad 2b-a^2 = 0,$$ $$ x^4: \quad 2d - 2ac +b^2 = 0,$$ $$x^2: \quad 2bd - c^2 = 4,$$ $$ x^0:\quad d^2 = 4.$$ Those equations have a solution $a=b=c=d=2$, and the coefficients of the odd powers of $x$ then agree automatically. Therefore $$x^8 + 4x^2 + 4 = (x^4 + 2x^3 + 2x^2 + 2x + 2) (x^4 - 2x^3 + 2x^2 - 2x + 2).$$
 
Back
Top