Factor completely. 9x^(6)-1 Answer:

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Factor completely.  9x^(6)-1 Answer:

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Recognize Difference of Squares: Step Title: Recognize the Difference of Squares Concise Step Description: Identify that the expression is a difference of squares, which can be factored into the product of a sum and difference. Step Calculation: Recognize that $9x^6$ is a perfect square, as is 1, so the expression $9x^6 - 1$ can be written as $(3x^3)^2 - 1^2$.Apply Squares Formula: Step Title: Apply the Difference of Squares Formula Concise Step Description: Use the difference of squares formula $a^2 - b^2 = (a + b)(a - b)$ to factor the expression. Step Calculation: Apply the formula with $a = 3x^3$ and $b = 1$ to get $(3x^3 + 1)(3x^3 - 1)$.Factor Further if Possible: Step Title: Factor Further if Possible Concise Step Description: Check if the resulting binomials can be factored further. Step Calculation: Notice that $3x^3 + 1$ cannot be factored further over the integers, but $3x^3 - 1$ is again a difference of squares, as it can be written as $(\sqrt[3]{3}x)^3 - 1^3$.Apply Cubes Formula: Step Title: Apply the Difference of Cubes Formula Concise Step Description: Use the difference of cubes formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ to factor $3x^3 - 1$. Step Calculation: Apply the formula with $a = \sqrt[3]{3}x$ and $b = 1$ to get $(\sqrt[3]{3}x - 1)((\sqrt[3]{3}x)^2 + (\sqrt[3]{3}x)(1) + 1^2)$.Simplify Resulting Expression: Step Title: Simplify the Resulting Expression Concise Step Description: Simplify the expression obtained from the difference of cubes formula. Step Calculation: The simplified form is $(\sqrt[3]{3}x - 1)(3x^2 + \sqrt[3]{3}x + 1)$.Combine All Factors: Step Title: Combine All Factors Concise Step Description: Combine all factors obtained from the previous steps to write the completely factored form of the original expression. Step Calculation: The completely factored form is $(3x^3 + 1)(\sqrt[3]{3}x - 1)(3x^2 + \sqrt[3]{3}x + 1)$.

Factor completely. $\newline$ $9 x^{6}-1$ $\newline$ Answer:

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Factor completely. $\newline$ $9 x^{6}-1$ $\newline$ Answer: