CGD of 2 polynomials

[Liene]

Hello!
I am trying to solve this excersise for a long time and can't get the right answer.
There is 2 polynomials:
{6x^3+7x^2-x-2, 2x^3+9x^2+7x-6}

Have to find common greatest divisor by using Gauss elimination method.
I started my solving with:
p1=p-3Q = 34x^2+20x+16
what next...
I saw that the main result should be 1, but can't get to that by solving that on paper.
Please help me someone!
Many thanks for patience to think about this!

 

[Bob Brown MSEE]

I get

20xx+22x-16=0

If you solve you will find one of the roots common wit P(x) and Q(x)

 

[HallsofIvy]

Hello!
I am trying to solve this excersise for a long time and can't get the right answer.
There is 2 polynomials:
{6x^3+7x^2-x-2, 2x^3+9x^2+7x-6}

Have to find common greatest divisor by using Gauss elimination method.

What, exactly is the "Gauss elimination method"? Any time you are required to use a specific method, you should always say what that method is.

I started my solving with:
p1=p-3Q = 34x^2+20x+16

This makes no sense because you have not told us what "p1", "p", and "Q" represent.
Assuming that the first polynomial, 6x^3+7x^2- x- 2, is "p" and that the second, 2x^3- 9x^2+ 7x- 6, is "Q" (I would have used "q") then p- 3q= (6x^3+ 7x^2- x- 2)- (6x^3- 27x^2+ 21x- 18)= 34x^2- 22x+ 16. (Note the "22" rather than "20".)

And, of course, if polynomial r divides both p and q, it must also divide ap+ bq for any a and b. In particular, a common factor of p and q must also be a factor of p- 3q. Since that is now quadratic, we can find its factors by solving the quadratic equation 34x^2- 22x+ 16= 0. The first thing I would do is divide by 2: 17x^2- 11x+ 8= 0.
You could solve that using the quadratic equation:
\(\displaystyle x= \frac{11\pm\sqrt{11^2- 4(17)(8)}}{2(17)}= \frac{11\pm\sqrt{121- 544}}{34}\)
but those are complex numbers. This polynomial cannot be factored in terms of real coefficients so the original polynomials have no common factor. The "greatest common divisor" is indeed "1".

what next...
I saw that the main result should be 1, but can't get to that by solving that on paper.
Please help me someone!
Many thanks for patience to think about this!

 

[soroban]

Hello, Liene!

\(\displaystyle \text{Find the greatest common divisor: }\:\begin{Bmatrix}P(x) &=& 6x^3 + 7x^2 - x - 2 \\ Q(x) &=& 2x^3 + 9x^2 + 7x - 6 \end{Bmatrix}\)

We must factor the two polynomials . . .


We find that \(\displaystyle P(\text{-}1) = 0\). . Hence, \(\displaystyle x+1\) is a factor of \(\displaystyle P(x).\)

Long division: .\(\displaystyle (6x^3 + 7x^2-x-2) \div (x+1) \;=\;6x^2 + x - 2\)

. . Hence: .\(\displaystyle P(x) \;=\;(x+1)\color{blue}{(2x-1)}(3x+2)\)


We find that \(\displaystyle Q(\text{-}2) = 0\) . Hence, \(\displaystyle x+2\) is a factor of \(\displaystyle Q(x).\)

Long division: .\(\displaystyle (2x^3 + 9x^2 + 7x - 6) \div (x + 2) \;=\;2x^2 + 5x - 3\)

. . Hence: .\(\displaystyle Q(x) \;=\;(x+2)(x+3)\color{blue}{(2x-1)}\)


Therefore, the GCD of \(\displaystyle P(x)\) and \(\displaystyle Q(x)\) is: \(\displaystyle \color{blue}{(2x-1)}\)