Find real solutions for p(b)

Question

I have a polynomial $p(b)=b^5 -b^4 -3b^3 -7b^2 -5b -3=0$. I don’t know how to solve that. Can somebody please explain how to factorize this step by step?

Answer 1

The 1,-1,3,-3 does not look right. For example, 1 cannot be a root, just add up the coefficients. One useful test is looking for repeat factors, find polynomial gcd of your polynomial and its formal derivative over the rationals. The outcome is $x^2 + x + 1,$ meaning that $ x^{5} - x^{4} - 3 x^{3} - 7 x^{2} - 5 x - 3$ is actually divisible by $(x^2 + x + 1)^2 = x^4 + 2 x^3 + 3 x^2 + 2x + 1,$ polynomial division gives

$$ \left( x^{5} - x^{4} - 3 x^{3} - 7 x^{2} - 5 x - 3 \right) = \left( x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1 \right) \cdot \left( x - 3 \right) $$

$$ \left( x^{5} - x^{4} - 3 x^{3} - 7 x^{2} - 5 x - 3 \right) = \left(x^2 + x + 1 \right)^2 \cdot \left( x - 3 \right) $$

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$$ \left( x^{5} - x^{4} - 3 x^{3} - 7 x^{2} - 5 x - 3 \right) $$

$$ \left( 5 x^{4} - 4 x^{3} - 9 x^{2} - 14 x - 5 \right) $$

$$ \left( x^{5} - x^{4} - 3 x^{3} - 7 x^{2} - 5 x - 3 \right) = \left( 5 x^{4} - 4 x^{3} - 9 x^{2} - 14 x - 5 \right) \cdot \color{magenta}{ \left( \frac{ 5 x - 1 }{ 25 } \right) } + \left( \frac{ - 34 x^{3} - 114 x^{2} - 114 x - 80 }{ 25 } \right) $$ $$ \left( 5 x^{4} - 4 x^{3} - 9 x^{2} - 14 x - 5 \right) = \left( \frac{ - 34 x^{3} - 114 x^{2} - 114 x - 80 }{ 25 } \right) \cdot \color{magenta}{ \left( \frac{ - 2125 x + 8825 }{ 578 } \right) } + \left( \frac{ 12675 x^{2} + 12675 x + 12675 }{ 289 } \right) $$ $$ \left( \frac{ - 34 x^{3} - 114 x^{2} - 114 x - 80 }{ 25 } \right) = \left( \frac{ 12675 x^{2} + 12675 x + 12675 }{ 289 } \right) \cdot \color{magenta}{ \left( \frac{ - 9826 x - 23120 }{ 316875 } \right) } + \left( 0 \right) $$ $$ \frac{ 0}{1} $$ $$ \frac{ 1}{0} $$ $$ \color{magenta}{ \left( \frac{ 5 x - 1 }{ 25 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ 5 x - 1 }{ 25 } \right) }{ \left( 1 \right) } $$ $$ \color{magenta}{ \left( \frac{ - 2125 x + 8825 }{ 578 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ - 425 x^{2} + 1850 x + 225 }{ 578 } \right) }{ \left( \frac{ - 2125 x + 8825 }{ 578 } \right) } $$ $$ \color{magenta}{ \left( \frac{ - 9826 x - 23120 }{ 316875 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ 289 x^{3} - 578 x^{2} - 578 x - 867 }{ 12675 } \right) }{ \left( \frac{ 1445 x^{2} - 2601 x - 1445 }{ 12675 } \right) } $$ $$ \left( x^{3} - 2 x^{2} - 2 x - 3 \right) \left( \frac{ - 85 x + 353 }{ 1014 } \right) - \left( 5 x^{2} - 9 x - 5 \right) \left( \frac{ - 17 x^{2} + 74 x + 9 }{ 1014 } \right) = \left( -1 \right) $$ $$ \left( x^{5} - x^{4} - 3 x^{3} - 7 x^{2} - 5 x - 3 \right) = \left( x^{3} - 2 x^{2} - 2 x - 3 \right) \cdot \color{magenta}{ \left( x^{2} + x + 1 \right) } + \left( 0 \right) $$ $$ \left( 5 x^{4} - 4 x^{3} - 9 x^{2} - 14 x - 5 \right) = \left( 5 x^{2} - 9 x - 5 \right) \cdot \color{magenta}{ \left( x^{2} + x + 1 \right) } + \left( 0 \right) $$ $$ \mbox{GCD} = \color{magenta}{ \left( x^{2} + x + 1 \right) } $$ $$ \left( x^{5} - x^{4} - 3 x^{3} - 7 x^{2} - 5 x - 3 \right) \left( \frac{ - 85 x + 353 }{ 1014 } \right) - \left( 5 x^{4} - 4 x^{3} - 9 x^{2} - 14 x - 5 \right) \left( \frac{ - 17 x^{2} + 74 x + 9 }{ 1014 } \right) = \left( - x^{2} - x - 1 \right) $$