Question

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which of the following factorizations is correct?
$x^3 + 64 = (x - 4)(x^2 + 4x - 16)$
$x^3 + 27 = (x - 4)(x^2 - 16)$
$x^3 - 64 = (x - 4)(x^2 + 4x + 16)$
$x^3 - 64 = (x - 4)(x^2 + 16)$

Explanation:

To determine the correct factorization, we use the sum and difference of cubes formulas. The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\) and the difference of cubes formula is \(a^3 - b^3=(a - b)(a^2 + ab + b^2)\).

Step 1: Analyze \(x^3 + 64\)

We can rewrite \(x^3 + 64\) as \(x^3+4^3\). Using the sum of cubes formula \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\) with \(a = x\) and \(b = 4\), we get \((x + 4)(x^2-4x + 16)\), which is not equal to \((x - 4)(x^2 + 4x - 16)\). So this option is incorrect.

Step 2: Analyze \(x^3 + 27\)

Rewrite \(x^3 + 27\) as \(x^3+3^3\). Using the sum of cubes formula \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\) with \(a = x\) and \(b = 3\), we get \((x + 3)(x^2-3x + 9)\). The given option is \((x - 4)(x^2 - 16)\), which is incorrect.

Step 3: Analyze \(x^3 - 64\)

Rewrite \(x^3 - 64\) as \(x^3-4^3\). Using the difference of cubes formula \(a^3 - b^3=(a - b)(a^2 + ab + b^2)\) with \(a = x\) and \(b = 4\), we get \((x - 4)(x^2+4x + 16)\), which matches the third option.

Step 4: Analyze \(x^3 - 64=(x - 4)(x^2 + 16)\)

If we expand \((x - 4)(x^2 + 16)=x^3+16x-4x^2 - 64=x^3-4x^2+16x - 64\), which is not equal to \(x^3 - 64\). So this option is incorrect.

Answer:

\(x^3 - 64 = (x - 4)(x^2 + 4x + 16)\) (the third option)