Pergunta

Question 10(Multiplo Choice Worth 1 points) (07.04 MC) Factor completely x^8-81 (x^4-9)(x^4+9) (x^2-3)(x^2+3)(x^4-9) (x^2-3)(x^2+3)(x^4+9) (x^2-3)(x^2-3)(x^4-9)
Solução

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GiovanniVeterano · Tutor por 10 anos
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To factor the expression \(x^8 - 81\) completely, we start by recognizing that it is a difference of squares. The difference of squares formula is given by:<br /><br />\[ a^2 - b^2 = (a - b)(a + b) \]<br /><br />In this case, we can rewrite \(x^8 - 81\) as:<br /><br />\[ x^8 - 81 = (x^4)^2 - 9^2 \]<br /><br />Now, applying the difference of squares formula:<br /><br />\[ (x^4)^2 - 9^2 = (x^4 - 9)(x^4 + 9) \]<br /><br />Next, we notice that \(x^4 - 9\) is also a difference of squares:<br /><br />\[ x^4 - 9 = (x^2)^2 - 3^2 = (x^2 - 3)(x^2 + 3) \]<br /><br />So, we can factor \(x^4 - 9\) further:<br /><br />\[ x^4 - 9 = (x^2 - 3)(x^2 + 3) \]<br /><br />Putting it all together, we have:<br /><br />\[ x^8 - 81 = (x^4 - 9)(x^4 + 9) = (x^2 - 3)(x^2 + 3)(x^4 + 9) \]<br /><br />Thus, the completely factored form of \(x^8 - 81\) is:<br /><br />\[ (x^2 - 3)(x^2 + 3)(x^4 + 9) \]<br /><br />Therefore, the correct answer is:<br /><br />\[ (x^2 - 3)(x^2 + 3)(x^4 + 9) \]
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