Factor completely: z^(9)+1 Answer:

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Factor completely:  z^(9)+1 Answer:

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Recognize Sum of Cubes: Step Title: Recognize the Sum of Two Cubes Concise Step Description: Identify that the expression is a sum of two cubes, which can be factored using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). Step Calculation: In this case, z^9 is (z^3)^3 and 1 is 1^3, so we can apply the sum of cubes formula. Step Output: Recognize that z^9 + 1 is a sum of two cubes.Apply Sum of Cubes Formula: Step Title: Apply the Sum of Two Cubes Formula Concise Step Description: Apply the sum of cubes formula to factor the expression. Step Calculation: Using the formula, we get (z^3 + 1)((z^3)^2 - z^3*1 + 1^2). Step Output: Factored form is (z^3 + 1)(z^6 - z^3 + 1).Check Further Factorization: Step Title: Check for Further Factorization Concise Step Description: Check if the factors obtained can be factored further. Step Calculation: The first factor z^3 + 1 is a sum of cubes again and can be factored further. The second factor z^6 - z^3 + 1 is not factorable over the real numbers. Step Output: Further factor the first term (z^3 + 1) using the sum of cubes formula.Factor First Term Further: Step Title: Factor the First Term Further Concise Step Description: Factor the first term (z^3 + 1) using the sum of cubes formula. Step Calculation: We have z^3 + 1^3, which factors to (z + 1)(z^2 - z*1 + 1^2). Step Output: Factored form is (z + 1)(z^2 - z + 1).Combine Factored Terms: Step Title: Combine the Factored Terms Concise Step Description: Combine the factored terms to get the completely factored expression. Step Calculation: The completely factored expression is (z + 1)(z^2 - z + 1)(z^6 - z^3 + 1). Step Output: The final factored form is (z + 1)(z^2 - z + 1)(z^6 - z^3 + 1).

Factor completely: $\newline$ $z^{9}+1$ $\newline$ Answer:

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