Questions: Select the correct answer. What is the factored form of 343+x^6 ? A. (7-x)(49+7x+x^2) B. (7+x)(49-7x+x^2) C. (7-x^2)(49+7x^2+x^4) D. (7+x^2)(49-7x^2+x^4)

Solution Steps

To factor the expression \(343 + x^6\), we recognize that \(343\) is a perfect cube, specifically \(7^3\). The expression can be rewritten as a sum of cubes: \(7^3 + (x^2)^3\). We can use the sum of cubes formula, which is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), where \(a = 7\) and \(b = x^2\).

1. Recognize the expression as a sum of cubes.
2. Apply the sum of cubes formula.

The expression \(343 + x^6\) can be rewritten as \(x^6 + 343\). Noticing that \(343\) is a perfect cube, we can express it as \(7^3\). Thus, we have:

\[ x^6 + 343 = x^6 + 7^3 \]

We can apply the sum of cubes formula, which states:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

In our case, let \(a = x^2\) and \(b = 7\). Therefore, we can factor the expression as follows:

\[ x^6 + 7^3 = (x^2 + 7)(x^4 - 7x^2 + 49) \]

Final Answer