To factor the expression \(5v^5 - 5v^4 + 6v^2 - 6v^3\), we can start by rearranging it slightly for easier factoring:
\[ 5v^5 - 5v^4 - 6v^3 + 6v^2 \]
Next, we can group the terms in pairs:
\[ (5v^5 - 5v^4) + (-6v^3 + 6v^2) \]
Now, we can factor out the greatest common factor from each group:
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From the first group \(5v^5 - 5v^4\), we can factor out \(5v^4\): \[ 5v^4(v - 1) \]
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From the second group \(-6v^3 + 6v^2\), we can factor out \(-6v^2\): \[ -6v^2(v - 1) \]
Now, we can write the expression as: \[ 5v^4(v - 1) - 6v^2(v - 1) \]
Since \( (v - 1) \) is a common factor, we can now factor it out: \[ (v - 1)(5v^4 - 6v^2) \]
Now we need to factor \(5v^4 - 6v^2\). We can factor out \(v^2\) from it:
\[ 5v^2(v^2 - \frac{6}{5}) \]
At this point, we notice that \(v^2 - \frac{6}{5}\) can be further factored:
\[ v^2 - \frac{6}{5} = v^2 - \left(\sqrt{\frac{6}{5}}\right)^2 \]
This is in the form of a difference of squares and can be factored as follows:
\[ v^2 - \left(\sqrt{\frac{6}{5}}\right)^2 = \left(v - \sqrt{\frac{6}{5}}\right)\left(v + \sqrt{\frac{6}{5}}\right) \]
Putting everything together, we have:
\[ 5v^2(v - \sqrt{\frac{6}{5}})(v + \sqrt{\frac{6}{5}})(v - 1) \]
Thus, the completely factored form of the expression \(5v^5 - 5v^4 + 6v^2 - 6v^3\) is:
\[ (v - 1)(5v^2)(v - \sqrt{\frac{6}{5}})(v + \sqrt{\frac{6}{5}}) \]