Question

Resolve into factors : x24 - y 24
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Answer 1

Answer:

24 will be common so the answer will be 24(x-y)=0

Answer 2

Answer:

  The equation $x^{24} -y^{24}$ is resolved. The factors of $x^{24} -y^{24}$ are $(x+y)$ $(x-y)$ $(x^{2} +y^{2} )$ $(x^{2} +y^{2} -xy)$ $(x^{2} +y^{2} +xy)$ $(x^{4}+y^{4} )$ $(x^{4} +y^{4} -x^{2} y^{2} )$ $(x^{8} +y^{8} -x^{4} y^{4} )$.

Explanation:

• Given that the function is $x^{24} -y^{24}$.

• We have to find the factors of this function, $x^{24} -y^{24}$.

• We know that; $(x^{a} )^{b}$ = $x^{ab}$, here the power value of each term is 24.

• Therefore, we can write; 24 = 12 × 2

• Then, $x^{24} -y^{24}$ = $(x^{12}) ^{2}$ - $(y^{12}) ^{2}$

• Now we can apply the equation; $a^{2}$ - $b^{2}$ = (a + b) (a - b)

∴ $x^{24} -y^{24}$ = $(x^{12} +y^{12})$ $(x^{12} -y^{12} )$

• Now the power value of each term is 12. Therefore we can rewrite 12 as; 12 = 6 × 2

∴ $(x^{12} +y^{12})$ $(x^{12} -y^{12} )$ = $(x^{12} +y^{12})$ $((x^{6} )^{2} -(y^{6}) ^{2} )$

• Again we can apply the equation $a^{2}$ - $b^{2}$ = (a + b) (a - b)

⇒ $(x^{12} +y^{12})$ $(x^{12} -y^{12} )$ = $(x^{12} +y^{12})$ $(x^{6} +y^{6} )$ $(x^{6} -y^{6} )$

• Now the power value of each term is 6. Therefore we can rewrite 6 as; 6 = 3 x 2

∴ $(x^{12} +y^{12})$ $(x^{12} -y^{12} )$ = $(x^{12} +y^{12})$ $(x^{6} +y^{6} )$ $((x^{3}) ^{2} -(y^{3}) ^{2} )$

• Again apply the equation $a^{2}$ - $b^{2}$ = (a + b) (a - b)

⇒ $(x^{12} +y^{12})$ $(x^{12} -y^{12} )$ = $(x^{12} +y^{12})$ $(x^{6} +y^{6} )$ $(x^{3} +y^{3} )$ $(x^{3} -y^{3} )$

• Now we can use the equation; $a^{3} -b^{3}$ = $(a-b)$ $(a^{2} +b^{2} +ab)$

⇒ $(x^{12} +y^{12})$ $(x^{6} +y^{6} )$ $(x^{3} +y^{3} )$ $(x-y)$ $(x^{2} +y^{2} +xy)$

• By using the equation; $a^{3} +b^{3}$ = $(a+b)$ $(a^{2} +b^{2} -ab)$

⇒ $(x^{12} +y^{12})$ $(x^{6} +y^{6} )$ $(x^{2} +y^{2} -xy)$ $(x+y)$ $(x^{2} +y^{2} +xy)$ $(x-y)$

∴ $x^{24} -y^{24}$ = $(x^{12} +y^{12})$ $(x^{6} +y^{6} )$ $(x^{2} +y^{2} -xy)$ $(x+y)$ $(x^{2} +y^{2} +xy)$ $(x-y)$

• Now, $x^{12} +y^{12}$ can be further factorize as; $(x^{4} )^{3} +(y^{4} )^{3}$. By using the equation; $a^{3} +b^{3}$ = $(a+b)$ $(a^{2} +b^{2} -ab)$

⇒ $(x^{4} )^{3} +(y^{4} )^{3}$ = $(x^{4}+y^{4} )$ $(x^{8} +y^{8} -x^{4} y^{4} )$

• Similarly, $x^{6} +y^{6}$ can be further factorize as; $(x^{2}) ^{3}$ + $(y^{2}) ^{3}$. By using the equation; $a^{3} +b^{3}$ = $(a+b)$ $(a^{2} +b^{2} -ab)$

⇒ $(x^{2}) ^{3}$ + $(y^{2}) ^{3}$ = $(x^{2} +y^{2} )$ $(x^{4} +y^{4} -x^{2} y^{2} )$

∴ $x^{24} -y^{24}$ = $(x^{4}+y^{4} )$ $(x^{8} +y^{8} -x^{4} y^{4} )$ $(x^{2} +y^{2} )$ $(x^{4} +y^{4} -x^{2} y^{2} )$ $(x^{2} +y^{2} -xy)$ $(x+y)$ $(x^{2} +y^{2} +xy)$ $(x-y)$

⇒ $x^{24} -y^{24}$ = $(x+y)$ $(x-y)$ $(x^{2} +y^{2} )$ $(x^{2} +y^{2} -xy)$ $(x^{2} +y^{2} +xy)$ $(x^{4}+y^{4} )$ $(x^{4} +y^{4} -x^{2} y^{2} )$ $(x^{8} +y^{8} -x^{4} y^{4} )$.

• Therefore, the factors of $x^{24} -y^{24}$ are $(x+y)$ $(x-y)$ $(x^{2} +y^{2} )$ $(x^{2} +y^{2} -xy)$ $(x^{2} +y^{2} +xy)$ $(x^{4}+y^{4} )$ $(x^{4} +y^{4} -x^{2} y^{2} )$ $(x^{8} +y^{8} -x^{4} y^{4} )$.
• Hence the equation $x^{24} -y^{24}$ is resolved.

To know more, go through the links;

brainly.in/question/7343350

brainly.in/question/18645339

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