Answer :
The polynomial [tex]\( x^3 + 64 \)[/tex] can be analyzed by recognizing special forms of polynomials. This polynomial is an example of a sum of cubes.
To see why this is the case, let's break it down further:
1. Identify the Terms: The expression [tex]\( x^3 + 64 \)[/tex] has two parts: [tex]\( x^3 \)[/tex] and 64.
2. Recognize Cubes: Notice that both terms can be expressed as cubes:
- [tex]\( x^3 \)[/tex] is indeed [tex]\( (x)^3 \)[/tex].
- 64 can be rewritten as [tex]\( 4^3 \)[/tex], because [tex]\( 4 \times 4 \times 4 = 64 \)[/tex].
3. Sum of Cubes Formula: A sum of cubes takes the general form [tex]\( A^3 + B^3 \)[/tex], where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are the bases of the cubes.
4. Apply the Formula: In the expression [tex]\( x^3 + 4^3 \)[/tex], we can identify:
- [tex]\( A = x \)[/tex]
- [tex]\( B = 4 \)[/tex]
Thus, the polynomial [tex]\( x^3 + 64 \)[/tex] matches the pattern of a sum of cubes, which further confirms it as a sum of cubes. Therefore, the correct classification for the polynomial [tex]\( x^3 + 64 \)[/tex] is a sum of cubes.
To see why this is the case, let's break it down further:
1. Identify the Terms: The expression [tex]\( x^3 + 64 \)[/tex] has two parts: [tex]\( x^3 \)[/tex] and 64.
2. Recognize Cubes: Notice that both terms can be expressed as cubes:
- [tex]\( x^3 \)[/tex] is indeed [tex]\( (x)^3 \)[/tex].
- 64 can be rewritten as [tex]\( 4^3 \)[/tex], because [tex]\( 4 \times 4 \times 4 = 64 \)[/tex].
3. Sum of Cubes Formula: A sum of cubes takes the general form [tex]\( A^3 + B^3 \)[/tex], where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are the bases of the cubes.
4. Apply the Formula: In the expression [tex]\( x^3 + 4^3 \)[/tex], we can identify:
- [tex]\( A = x \)[/tex]
- [tex]\( B = 4 \)[/tex]
Thus, the polynomial [tex]\( x^3 + 64 \)[/tex] matches the pattern of a sum of cubes, which further confirms it as a sum of cubes. Therefore, the correct classification for the polynomial [tex]\( x^3 + 64 \)[/tex] is a sum of cubes.
