Factor Using The Identity:$ A^3 + B^3 = (a + B)(a^2 - Ab + B^2) }$Given { Z^3 + 8 $ $Choose The Correct Factorization:A. { (z + 8)(z^2 - 8z + 64)$}$B. { (z + 2)(z^2 - 2z + 4)$}$C. { (z + 2)^3$}$D.

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Introduction

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In mathematics, factorization is a fundamental concept that plays a crucial role in solving equations and inequalities. One of the most commonly used factorization techniques is the sum of cubes formula, which states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. In this article, we will explore how to use this identity to factorize expressions, with a focus on the given problem: $z^3 + 8$.

Understanding the Sum of Cubes Formula

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The sum of cubes formula is a powerful tool for factorizing expressions of the form $a^3 + b^3$. This formula can be derived by using the difference of cubes formula, which states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. By adding $2b^3$ to both sides of this equation, we get $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Applying the Sum of Cubes Formula to the Given Problem

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Now that we have a solid understanding of the sum of cubes formula, let's apply it to the given problem: $z^3 + 8$. We can see that this expression is in the form of $a^3 + b^3$, where $a = z$ and $b = 2$. Therefore, we can use the sum of cubes formula to factorize this expression.

Step 1: Identify the Values of a and b

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In the given problem, $a = z$ and $b = 2$. These values will be used to substitute into the sum of cubes formula.

Step 2: Substitute the Values of a and b into the Sum of Cubes Formula

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Substituting $a = z$ and $b = 2$ into the sum of cubes formula, we get:

$$z^3 + 2^3 = (z + 2)(z^2 - z(2) + 2^2)$$

Step 3: Simplify the Expression

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Simplifying the expression, we get:

$$(z + 2)(z^2 - 2z + 4)$$

Conclusion

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Based on the steps outlined above, we can conclude that the correct factorization of $z^3 + 8$ is:

$$(z + 2)(z^2 - 2z + 4)$$

This is option B in the given problem.

Discussion

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In this article, we have demonstrated how to use the sum of cubes formula to factorize expressions. We have applied this formula to the given problem: $z^3 + 8$, and have shown that the correct factorization is $(z + 2)(z^2 - 2z + 4)$. This result is consistent with the sum of cubes formula, which states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Common Mistakes to Avoid

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When using the sum of cubes formula, there are several common mistakes to avoid. These include:

• Incorrectly identifying the values of a and b: Make sure to carefully identify the values of a and b in the given problem.
• Failing to substitute the values of a and b into the sum of cubes formula: Make sure to substitute the values of a and b into the sum of cubes formula.
• Not simplifying the expression: Make sure to simplify the expression after substituting the values of a and b into the sum of cubes formula.

Conclusion

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In conclusion, the sum of cubes formula is a powerful tool for factorizing expressions. By understanding this formula and applying it to the given problem, we have demonstrated how to factorize expressions of the form $a^3 + b^3$. We have also highlighted common mistakes to avoid when using the sum of cubes formula. With practice and patience, you will become proficient in using this formula to factorize expressions.

Final Answer

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The final answer is option B: $(z + 2)(z^2 - 2z + 4)$.

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Q&A: Frequently Asked Questions

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Q: What is the sum of cubes formula?

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A: The sum of cubes formula is a mathematical formula that states $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. This formula can be used to factorize expressions of the form $a^3 + b^3$.

Q: How do I apply the sum of cubes formula to a given problem?

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A: To apply the sum of cubes formula to a given problem, you need to identify the values of a and b in the expression. Then, substitute these values into the sum of cubes formula and simplify the expression.

Q: What are some common mistakes to avoid when using the sum of cubes formula?

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A: Some common mistakes to avoid when using the sum of cubes formula include:

• Incorrectly identifying the values of a and b: Make sure to carefully identify the values of a and b in the given problem.
• Failing to substitute the values of a and b into the sum of cubes formula: Make sure to substitute the values of a and b into the sum of cubes formula.
• Not simplifying the expression: Make sure to simplify the expression after substituting the values of a and b into the sum of cubes formula.

Q: Can I use the sum of cubes formula to factorize expressions of the form $a^3 - b^3$?

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A: No, the sum of cubes formula is only applicable to expressions of the form $a^3 + b^3$. For expressions of the form $a^3 - b^3$, you need to use the difference of cubes formula, which states $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Q: How do I factorize expressions of the form $a^3 - b^3$?

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A: To factorize expressions of the form $a^3 - b^3$, you need to use the difference of cubes formula. This involves identifying the values of a and b in the expression, substituting these values into the difference of cubes formula, and simplifying the expression.

Q: Can I use the sum of cubes formula to factorize expressions of the form $a^3 + b^3 + c^3$?

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A: No, the sum of cubes formula is only applicable to expressions of the form $a^3 + b^3$. For expressions of the form $a^3 + b^3 + c^3$, you need to use the sum of cubes formula twice, or use a different factorization technique.

Q: How do I factorize expressions of the form $a^3 + b^3 + c^3$?

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A: To factorize expressions of the form $a^3 + b^3 + c^3$, you need to use the sum of cubes formula twice, or use a different factorization technique. This involves identifying the values of a, b, and c in the expression, substituting these values into the sum of cubes formula, and simplifying the expression.

Conclusion

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In conclusion, the sum of cubes formula is a powerful tool for factorizing expressions of the form $a^3 + b^3$. By understanding this formula and applying it to given problems, you can factorize expressions with ease. We have also discussed common mistakes to avoid when using the sum of cubes formula, and provided guidance on how to factorize expressions of the form $a^3 - b^3$ and $a^3 + b^3 + c^3$.

Final Answer

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The final answer is option B: $(z + 2)(z^2 - 2z + 4)$.