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Math Help - How to Factorise This?

  1. #1
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    How to Factorise This?

    x^5+x^3+x

    Answer: x(1-x+x^2)(1+x+x^2)
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  2. #2
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    Quote Originally Posted by cloud5 View Post
    x^5+x^3+x

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


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  3. #3
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    Quote Originally Posted by yeongil View Post
    \begin{aligned}<br />
x^5+x^3+x &= x(x^4 + x^2 + 1) \\<br />
&= x(x^4\;{\color{red}+\;2x^2} + 1\;{\color{red}-\;x^2}) \\<br />
&= x[(x^2 + 1)^2 - x^2] \\<br />
&= x[(x^2 + 1) + x][(x^2 + 1) - x] \\<br />
&= x(x^2 + x + 1)(x^2 - x + 1)<br />
 \end{aligned}


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