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Factorize the quintic polynomial $P(x) = x^5 + 2x^4 -x -2$ into a product of irreducible linear and quadratic factors what are the roots of $p$?

Hi, i got this question in a worksheet but my text book doesn't have anything about quintic polynomials in it. if anybody could help me that would be great!

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  • $\begingroup$ use the rational root theorem $\endgroup$ – Dr. Sonnhard Graubner Mar 4 '17 at 7:41
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    $\begingroup$ $x^5-x+2(x^4-1)=x(x^4-1)+2(x^4-1)=(x+2)(x^4-1)=\cdots$ $\endgroup$ – mathlove Mar 4 '17 at 7:55
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Hint: Divisors of $-2$ are $$1,-1,-2,2$$ and you will find all roots if they are integers

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  • $\begingroup$ "... if they are integers". We have been fortunate here that all the roots are integers. $\endgroup$ – Jean Marie Mar 4 '17 at 8:55
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let us see the root of the $P(x)=0$. first try the factor of $\frac{-2}{1}=-2$, if $x=1,-1,-2$, the $P(x)=0$, so $P(x)$ must have factor $(x-1)(x+1)(x+2)$.

then use division, $\frac{P(x)}{(x-1)(x+1)(x+2)} = x^2+1$.

so $P(x) = (x-1)(x+1)(x+2)(x^2+1)$.

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