Select The Correct Answer.What Is The Factored Form Of 343 + X 6 343+x^6 343 + X 6 ?A. ( 7 − X ) ( 49 + 7 X + X 2 (7-x)(49+7x+x^2 ( 7 − X ) ( 49 + 7 X + X 2 ] B. ( 7 + X ) ( 49 − 7 X + X 2 (7+x)(49-7x+x^2 ( 7 + X ) ( 49 − 7 X + X 2 ] C. ( 7 − X 2 ) ( 49 + 7 X 2 + X 4 (7-x^2)(49+7x^2+x^4 ( 7 − X 2 ) ( 49 + 7 X 2 + X 4 ] D. ( 7 + X 2 ) ( 49 − 7 X 2 + X 4 (7+x^2)(49-7x^2+x^4 ( 7 + X 2 ) ( 49 − 7 X 2 + X 4 ]

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Understanding the Problem

The problem requires us to find the factored form of the expression $343+x^6$. To approach this problem, we need to understand the concept of factoring and how to apply it to the given expression. Factoring involves expressing an algebraic expression as a product of simpler expressions, called factors.

Identifying the Type of Expression

The given expression $343+x^6$ is a polynomial expression, which can be factored using various techniques. We need to identify the type of polynomial expression and choose the appropriate factoring technique.

Recognizing the Difference of Squares

Upon examining the expression, we notice that it can be written as a sum of two squares: $343+x^6 = 49^2 + x^6$. This suggests that we can use the difference of squares formula to factor the expression.

Applying the Difference of Squares Formula

The difference of squares formula states that $a^2 - b^2 = (a - b)(a + b)$. We can rewrite the expression $343+x^6$ as $(49)^2 + (x^3)^2$, which is a sum of two squares. This allows us to apply the difference of squares formula.

Factoring the Expression

Using the difference of squares formula, we can factor the expression $343+x^6$ as follows:

$343+x^6 = (49)^2 + (x^3)^2$ $= (49 + x^3)(49 - x^3)$

Simplifying the Expression

We can simplify the expression further by recognizing that $49 - x^3$ is a difference of squares. This allows us to apply the difference of squares formula again.

$49 - x^3 = (7 - x)(7 + x)$

Factoring the Expression Completely

Substituting the simplified expression back into the factored form, we get:

$343+x^6 = (49 + x^3)(7 - x)(7 + x)$

Comparing with the Answer Choices

We can now compare the factored form of the expression with the answer choices. The correct answer is:

A. $(7-x)(49+7x+x^2)$

This answer choice matches the factored form of the expression that we obtained using the difference of squares formula.

Conclusion

In conclusion, we have successfully factored the expression $343+x^6$ using the difference of squares formula. The correct answer is A. $(7-x)(49+7x+x^2)$.

Key Takeaways

• The difference of squares formula can be used to factor expressions of the form $a^2 - b^2$.
• The sum of two squares can be rewritten as a difference of squares using the formula $(a^2 + b^2) = (a + ib)(a - ib)$.
• Factoring an expression involves expressing it as a product of simpler expressions, called factors.

Final Answer

The final answer is A. $(7-x)(49+7x+x^2)$.

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Understanding the Problem

The problem requires us to find the factored form of the expression $343+x^6$. To approach this problem, we need to understand the concept of factoring and how to apply it to the given expression. Factoring involves expressing an algebraic expression as a product of simpler expressions, called factors.

Q: What is the factored form of $343+x^6$?

A: The factored form of $343+x^6$ is $(7-x)(49+7x+x^2)$.

Q: How do we factor the expression $343+x^6$?

A: We can factor the expression $343+x^6$ using the difference of squares formula. The difference of squares formula states that $a^2 - b^2 = (a - b)(a + b)$. We can rewrite the expression $343+x^6$ as $(49)^2 + (x^3)^2$, which is a sum of two squares. This allows us to apply the difference of squares formula.

Q: What is the difference of squares formula?

A: The difference of squares formula states that $a^2 - b^2 = (a - b)(a + b)$. This formula can be used to factor expressions of the form $a^2 - b^2$.

Q: How do we apply the difference of squares formula to the expression $343+x^6$?

A: We can rewrite the expression $343+x^6$ as $(49)^2 + (x^3)^2$, which is a sum of two squares. This allows us to apply the difference of squares formula. Using the formula, we can factor the expression as follows:

$343+x^6 = (49)^2 + (x^3)^2$ $= (49 + x^3)(49 - x^3)$

Q: What is the next step in factoring the expression $343+x^6$?

A: We can simplify the expression further by recognizing that $49 - x^3$ is a difference of squares. This allows us to apply the difference of squares formula again.

Q: How do we simplify the expression $49 - x^3$?

A: We can simplify the expression $49 - x^3$ by recognizing that it is a difference of squares. Using the difference of squares formula, we can factor the expression as follows:

$49 - x^3 = (7 - x)(7 + x)$

Q: What is the final factored form of the expression $343+x^6$?

A: The final factored form of the expression $343+x^6$ is $(7-x)(49+7x+x^2)$.

Q: Why is the answer choice A the correct answer?

A: The answer choice A is the correct answer because it matches the factored form of the expression that we obtained using the difference of squares formula.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are:

• The difference of squares formula can be used to factor expressions of the form $a^2 - b^2$.
• The sum of two squares can be rewritten as a difference of squares using the formula $(a^2 + b^2) = (a + ib)(a - ib)$.
• Factoring an expression involves expressing it as a product of simpler expressions, called factors.

Q: What is the final answer to the problem?

A: The final answer to the problem is A. $(7-x)(49+7x+x^2)$.