Verified Answer :

  • The constant term 64 is equal to \(4^3\), allowing the polynomial \(x^3 + 64\) to be rewritten as \(x^3 + 4^3\).
  • This form corresponds directly to the identity for a sum of cubes, which is \(a^3 + b^3\).
  • Comparing this to the given options shows that the polynomial fits the definition of a sum of cubes perfectly.
  • The correct answer is \(\boxed{a}\).

Explanation

  1. Analysis of the Polynomial Structure

We are tasked with identifying the mathematical structure of the polynomial \(x^3 + 64\). To do this, we analyze the expression by comparing it to the standard forms listed in the options. A polynomial is composed of variables and coefficients, and specific algebraic identities help us classify them based on their structure. The expression given is \(x^3 + 64\). We first check if the constant term, 64, can be written as a power of another number. Since \(4 \times 4 \times 4 = 64\), we can rewrite the constant as \(4^3\). Therefore, the polynomial can be expressed as \(x^3 + 4^3\).

  1. Evaluating the Options

Now, let's examine the definitions of the provided options:

ASum of cubes: This is an algebraic expression of the form \(a^3 + b^3\), which factors as \((a+b)(a^2 - ab + b^2)\). Since we have expressed \(x^3 + 64\) as \(x^3 + 4^3\), it perfectly matches this definition with \(a = x\) and \(b = 4\).
BDifference of cubes: This is an expression of the form \(a^3 - b^3\). Our expression involves an addition sign, so this is not correct.
CDifference of squares: This is an expression of the form \(a^2 - b^2\). Our polynomial is cubic and involves a sum, not a difference of squares.
DPerfect-square trinomial: This is an expression of the form \((a \pm b)^2 = a^2 \pm 2ab + b^2\), which consists of three terms. Our expression is a binomial (two terms), not a trinomial.

Comparing our expression \(x^3 + 4^3\) with these definitions, it is clear that it fits the definition of a sum of cubes.

  1. Conclusion

Based on the identification that \(x^3 + 64 = x^3 + 4^3\), the polynomial is a classic example of a sum of cubes. Therefore, the correct option is (a).

Examples

Understanding algebraic structures like the "sum of cubes" is vital for simplifying expressions in physics and engineering. For example, when calculating the volume of combined objects or analyzing rotational motion in mechanics, you might encounter formulas where terms represent dimensions cubed. Recognizing these patterns allows you to factor them quickly, such as \(x^3 + 64 = (x+4)(x^2 - 4x + 16)\), simplifying complex calculations into manageable products. This is a foundational skill for solving equations efficiently.

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