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Find the Bezout coefficients for $a(x)$ and $b(x)$:

$a(x)=3x^4-4x^3-11x^2+4x+9, b(x)=3x^3+5x^2+x-1$

I find the greatest common divisor:

1) $\frac{(3x^4-4x^3-11x^2+4x+9)}{(3x^3+5x^2+x-1)} = x-3$. Remainder of the division: 3x^2+8x+6

2) $\frac{(3x^3+5x^2+x-1)}{(3x^2+8x+6)} = x-1$. Remainder of the division: 3x+5

3) $\frac{(x^2+8x+6)}{(3x+5)} = x+1$. Remainder of the division: $1$

The greatest common divisor: $1$

I find the Bezout coefficients with the help of the extended Euclidean algorithm:

$1\mid 0$

$0\mid 1$

$1\mid -x+3$

$-x+1\mid -x^2+4x-2$

$x^2\mid x^3-4x^2+2x$

I'm doing a check. It should work like this: $(x^2)*a(x) + (x^3-4x^2+2x)*b(x) = 1$

But here an error, the last line $(x^2\mid x^3-4x^2+2x)$ was not calculated correctly. Help me please.

1ENİGMA1
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GeorgeSV
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1 Answers1

1

I got

$$ \left( 3 x^{4} - 4 x^{3} - 11 x^{2} + 4 x + 9 \right) $$

$$ \left( 3 x^{3} + 5 x^{2} + x - 1 \right) $$

$$ \left( 3 x^{4} - 4 x^{3} - 11 x^{2} + 4 x + 9 \right) = \left( 3 x^{3} + 5 x^{2} + x - 1 \right) \cdot \color{magenta}{ \left( x - 3 \right) } + \left( 3 x^{2} + 8 x + 6 \right) $$ $$ \left( 3 x^{3} + 5 x^{2} + x - 1 \right) = \left( 3 x^{2} + 8 x + 6 \right) \cdot \color{magenta}{ \left( x - 1 \right) } + \left( 3 x + 5 \right) $$ $$ \left( 3 x^{2} + 8 x + 6 \right) = \left( 3 x + 5 \right) \cdot \color{magenta}{ \left( x + 1 \right) } + \left( 1 \right) $$ $$ \left( 3 x + 5 \right) = \left( 1 \right) \cdot \color{magenta}{ \left( 3 x + 5 \right) } + \left( 0 \right) $$ $$ \frac{ 0}{1} $$ $$ \frac{ 1}{0} $$ $$ \color{magenta}{ \left( x - 3 \right) } \Longrightarrow \Longrightarrow \frac{ \left( x - 3 \right) }{ \left( 1 \right) } $$ $$ \color{magenta}{ \left( x - 1 \right) } \Longrightarrow \Longrightarrow \frac{ \left( x^{2} - 4 x + 4 \right) }{ \left( x - 1 \right) } $$ $$ \color{magenta}{ \left( x + 1 \right) } \Longrightarrow \Longrightarrow \frac{ \left( x^{3} - 3 x^{2} + x + 1 \right) }{ \left( x^{2} \right) } $$ $$ \color{magenta}{ \left( 3 x + 5 \right) } \Longrightarrow \Longrightarrow \frac{ \left( 3 x^{4} - 4 x^{3} - 11 x^{2} + 4 x + 9 \right) }{ \left( 3 x^{3} + 5 x^{2} + x - 1 \right) } $$ $$ \left( 3 x^{4} - 4 x^{3} - 11 x^{2} + 4 x + 9 \right) \left( x^{2} \right) - \left( 3 x^{3} + 5 x^{2} + x - 1 \right) \left( x^{3} - 3 x^{2} + x + 1 \right) = \left( 1 \right) $$

Will Jagy
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