Question

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Explanation:

Identify the algebraic identities

We use the given formulas for the sum and difference of cubes:

• Sum of cubes: \((a + b)(a^2 - ab + b^2) = a^3 + b^3\)
• Difference of cubes: \((a - b)(a^2 + ab + b^2) = a^3 - b^3\)

Analyze options with linear factor \((x - 4)\) or \((x + 4)\)

Using Polynomial Operations, we test \(a = x\) and \(b = 4\):

• For \((x - 4)\), we need \(a^2 + ab + b^2 = x^2 + 4x + 16\).
• Option 1: \((x - 4)(x^2 + 4x - 16)\) does not match.
• For \((x + 4)\), we need \(a^2 - ab + b^2 = x^2 - 4x + 16\).
• Option 5: \((x + 4)(x^2 - 4x + 16)\) matches perfectly.
• Option 6: \((x + 4)(x^2 + 4x + 16)\) does not match.

Analyze options with linear factor \((x - 1)\) or \((x + 1)\)

Using Polynomial Multiplication, we test \(a = x\) and \(b = 1\):

• For \((x - 1)\), we need \(a^2 + ab + b^2 = x^2 + x + 1\).
• Option 2: \((x - 1)(x^2 - x + 1)\) does not match.
• Option 3: \((x - 1)(x^2 + x + 1)\) matches perfectly.
• For \((x + 1)\), we need \(a^2 - ab + b^2 = x^2 - x + 1\).
• Option 4: \((x + 1)(x^2 + x - 1)\) does not match.

Select the correct options

The products that result in a sum or difference of cubes are:

• \((x - 1)(x^2 + x + 1) = x^3 - 1\)
• \((x + 4)(x^2 - 4x + 16) = x^3 + 64\)

Answer:

• (A) \((x - 4)(x^2 + 4x - 16)\)
• (B) \((x - 1)(x^2 - x + 1)\)
• (C) \((x - 1)(x^2 + x + 1)\) (Correct answer)
• (D) \((x + 1)(x^2 + x - 1)\)
• (E) \((x + 4)(x^2 - 4x + 16)\) (Correct answer)
• (F) \((x + 4)(x^2 + 4x + 16)\)