Factor polynomial $$ c^{10}-1024m^{10} $$

$$ c^{10}-1024m^{10} = (c+2m)(c^{4}-2c^{3}m+4c^{2}m^{2}-8cm^{3}+16m^{4})(c-2m)(c^{4}+2c^{3}m+4c^{2}m^{2}+8cm^{3}+16m^{4}) $$

Step 1 :

Rewrite $ c^10-1024m^10 $ as:

$$ \color{blue}{ c^10-1024m^10 = (c^5)^2 - (32m^5)^2 } $$

Now we can apply the difference of squares formula.

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

After putting $ I = c^5 $ and $ II = 32m^5 $ , we have:

$$ c^10-1024m^10 = (c^5)^2 - (32m^5)^2 = ( c^5-32m^5 ) ( c^5+32m^5 ) $$

Step 2 :

To factor $ c^{5}+32m^{5} $ 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 = c $ and $ II = 2m $ , we have:

$$ c^{5}+32m^{5} = ( c+2m ) ( c^{4}-2c^{3}m+4c^{2}m^{2}-8cm^{3}+16m^{4} ) $$

Step 3 :

To factor $ c^{5}-32m^{5} $ 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 = c $ and $ II = 2m $ , we have:

$$ c^{5}-32m^{5} = ( c-2m ) ( c^{4}+2c^{3}m+4c^{2}m^{2}+8cm^{3}+16m^{4} ) $$

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