Highest common factor of two polynomials

Question

I have $$f(x) = x^3-2x^2-5x+6,\quad g(x)=x^2-2x-3$$

Then $f(x) = x(x^2-2x-3) +(-2x+6)$

So $hcf(f(x),g(x))=hcf((x^2-2x-3),(-2x+6))$

$x^2-2x-3=(-\frac{1}{2}x-\frac{1}{2})(-2x+6)+0$

So $hcf(f(x),g(x))=hcf((-2x+6),(0)) = 0$

Is this correct? If yes, then is the $0$ representing polynomials of degree $0$ or polynomials $h(x)$ such that $h(x)=0$?

Answer 1

When you get to zero, the highest common factor is the one you had just before. Here $2x-6=2(x-3)$ and $x-3$ is the highest common factor of the first polynomials.

Answer 2

The two polynomials satisfy $gcd(f,g)=1$, because $g(x)=(x+1)(x-3)$, and both $-1$ and $3$ are not a root of $f(x)=x^3-3x^2-5x+6$, i.e., both $(x+1)$ and $(x-3)$ are not a factor of $f$. So $1$ is the highest common factor.

Edit: The new polynomial now is $f(x)=x^3-2x^2-5x+6$. This changes the answer to $gcd(f,g)=x-3$.

Answer 3

$$ \left( x^{3} - 2 x^{2} - 5 x + 6 \right) $$

$$ \left( x^{2} - 2 x - 3 \right) $$

$$ \left( x^{3} - 2 x^{2} - 5 x + 6 \right) = \left( x^{2} - 2 x - 3 \right) \cdot \color{magenta}{ \left( x \right) } + \left( - 2 x + 6 \right) $$ $$ \left( x^{2} - 2 x - 3 \right) = \left( - 2 x + 6 \right) \cdot \color{magenta}{ \left( \frac{ - x - 1 }{ 2 } \right) } + \left( 0 \right) $$ $$ \frac{ 0}{1} $$ $$ \frac{ 1}{0} $$ $$ \color{magenta}{ \left( x \right) } \Longrightarrow \Longrightarrow \frac{ \left( x \right) }{ \left( 1 \right) } $$ $$ \color{magenta}{ \left( \frac{ - x - 1 }{ 2 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ - x^{2} - x + 2 }{ 2 } \right) }{ \left( \frac{ - x - 1 }{ 2 } \right) } $$ $$ \left( x^{2} + x - 2 \right) \left( \frac{ 1}{2 } \right) - \left( x + 1 \right) \left( \frac{ x }{ 2 } \right) = \left( -1 \right) $$

$$ \left( x^{3} - 2 x^{2} - 5 x + 6 \right) = \left( x^{2} + x - 2 \right) \cdot \color{magenta}{ \left( x - 3 \right) } + \left( 0 \right) $$ $$ \left( x^{2} - 2 x - 3 \right) = \left( x + 1 \right) \cdot \color{magenta}{ \left( x - 3 \right) } + \left( 0 \right) $$ $$ \mbox{GCD} = \color{magenta}{ \left( x - 3 \right) } $$ $$ \left( x^{3} - 2 x^{2} - 5 x + 6 \right) \left( \frac{ 1}{2 } \right) - \left( x^{2} - 2 x - 3 \right) \left( \frac{ x }{ 2 } \right) = \left( - x + 3 \right) $$