GCD of Polynomials: Is 1 Always the Solution?

In summary, the GCD of two polynomials, x^2+x+c and (x-a)^2+(x-a)+c, will always be 1 since they do not share any common roots. This is also true for any two polynomials of the form a(x-t)^2+b(x-t)+c and ax^2+bx+c. However, the example x2 + 3x + 2 = 0 and (x + 1)2 + 3(x + 1) + 2 = 0 shows that this is not always the case.
  • #1
tgt
522
2
Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?
 
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  • #2
tgt said:
Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?

Hi tgt! :smile:

Hint: if the roots of the first one are p and q, what are the roots of the second one? :wink:
 
  • #3
So it seems that two polynomials have nonzero GCD when there is at leaste one root shared between the two. So any two polynomials of the form a(x-t)^2+b(x-t)+c and ax^2+bx+c must have GCD 1 since they wouldn't have any roots shared between them.
 
  • #4
tgt said:
So it seems that two polynomials have nonzero GCD when there is at leaste one root shared between the two. So any two polynomials of the form a(x-t)^2+b(x-t)+c and ax^2+bx+c must have GCD 1 since they wouldn't have any roots shared between them.

What about x2 + 3x + 2 = 0 and (x + 1)2 + 3(x + 1) + 2 = 0? :rolleyes:
 
  • #5
tiny-tim said:
What about x2 + 3x + 2 = 0 and (x + 1)2 + 3(x + 1) + 2 = 0? :rolleyes:

ok, I wasn't thinkng clearly at the time.
 

1. What is the GCD of polynomials?

The GCD (Greatest Common Divisor) of polynomials is the highest degree polynomial that can divide evenly into both polynomials with no remainder. It is also known as the highest common factor.

2. Is 1 always the solution for the GCD of polynomials?

No, 1 is not always the solution for the GCD of polynomials. It depends on the polynomials being evaluated. In some cases, the GCD may be a different constant or even a polynomial.

3. How do you find the GCD of polynomials?

The GCD of polynomials can be found using the Euclidean algorithm, which involves dividing the larger polynomial by the smaller one until the remainder is zero. The last non-zero remainder is the GCD of the two polynomials.

4. Can the GCD of polynomials be negative?

Yes, the GCD of polynomials can be negative. The GCD is a factor of both polynomials, so it can be positive or negative depending on the factors of the polynomials being evaluated.

5. What is the significance of finding the GCD of polynomials?

Finding the GCD of polynomials can help simplify and factorize complex polynomials. It can also be used in various mathematical operations, such as finding common denominators or solving equations.

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