Factor polynomial $$ v^{10}-1 $$

$$ v^{10}-1 = ( v-1 )( v^{4}+v^{3}+v^{2}+v+1 )( v+1 )( v^{4}-v^{3}+v^{2}-v+1 ) $$

Step 1 :

Rewrite $ v^{10}-1 $ as:

$$ v^{10}-1 = (v^{5})^2 - (1)^2 $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = v^{5} $ and $ II = 1 $ , we have:

$$ v^{10}-1 = (v^{5})^2 - (1)^2 = ( v^{5}-1 ) ( v^{5}+1 ) $$

Step 2 :

To factor $ v^{5}-1 $ we can use formula:

$$ I^5 - II^5 = (I - II)(I^4 + I^3 \cdot II + I^2 \cdot II^2 + I \cdot II^3 + II^4) $$

After putting $ I = v $ and $ II = 1 $ , we have:

$$ v^{5}-1 = ( v-1 ) ( v^{4}+v^{3}+v^{2}+v+1 ) $$

Step 3 :

To factor $ v^{5}+1 $ we can use formula:

$$ I^5 + II^5 = (I + II)(I^4 - I^3 \cdot II + I^2 \cdot II^2 - I \cdot II^3 + II^4) $$

After putting $ I = v $ and $ II = 1 $ , we have:

$$ v^{5}+1 = ( v+1 ) ( v^{4}-v^{3}+v^{2}-v+1 ) $$

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