# Thread: How to Factorise This?

1. ## How to Factorise This?

$x^5+x^3+x$

Answer: $x(1-x+x^2)(1+x+x^2)$

2. Originally Posted by cloud5
$x^5+x^3+x$

Answer: $x(1-x+x^2)(1+x+x^2)$
\begin{aligned}
x^5+x^3+x &= x(x^4 + x^2 + 1) \\
&= x(x^4\;{\color{red}+\;2x^2} + 1\;{\color{red}-\;x^2}) \\
&= x[(x^2 + 1)^2 - x^2] \\
&= x[(x^2 + 1) + x][(x^2 + 1) - x] \\
&= x(x^2 + x + 1)(x^2 - x + 1)
\end{aligned}

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3. Originally Posted by yeongil
\begin{aligned}
x^5+x^3+x &= x(x^4 + x^2 + 1) \\
&= x(x^4\;{\color{red}+\;2x^2} + 1\;{\color{red}-\;x^2}) \\
&= x[(x^2 + 1)^2 - x^2] \\
&= x[(x^2 + 1) + x][(x^2 + 1) - x] \\
&= x(x^2 + x + 1)(x^2 - x + 1)
\end{aligned}

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How did you get the red colored numbers? I don't understand...

4. Originally Posted by cloud5
How did you get the red colored numbers? I don't understand...
\begin{aligned}
x^5+x^3+x &= x(x^4 + {\color{blue}x^2} + 1) \\
&= x(x^4\;{\color{red}+\;2x^2} + 1\;{\color{red}-\;x^2}) \\
\end{aligned}

...
Do you see the x-squared term in the previous step that I colored in blue? I replaced that with $2x^2 - x^2$, which is in red and separated. After all, $x^2 = 2x^2 - x^2$.

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