Factor polynomial $$ x^{15}-1 $$
Answer
$$ x^{15}-1 = ( x-1 )( x^{4}+x^{3}+x^{2}+x+1 )( x^{10}+x^{5}+1 ) $$
Explanation
Step 1 :
To factor $ x^{15}-1 $ we can use difference of cubes formula:
$$ I^3 - II^3 = (I - II)(I^2 + I \cdot II + II^2) $$After putting $ I = x^{5} $ and $ II = 1 $ , we have:
$$ x^{15}-1 = ( x^{5}-1 ) ( x^{10}+x^{5}+1 ) $$Step 2 :
To factor $ x^{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 = x $ and $ II = 1 $ , we have:
$$ x^{5}-1 = ( x-1 ) ( x^{4}+x^{3}+x^{2}+x+1 ) $$This page was created using
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