To determine whether each of the given expressions is a sum or difference of cubes, we need to check if they can be written in the forms:
1. [tex]\(a^3 + b^3\)[/tex] (sum of cubes)
2. [tex]\(a^3 - b^3\)[/tex] (difference of cubes)
The given expressions are:
1. [tex]\(64x^3 - 216\)[/tex]
2. [tex]\(8x^9 + 27\)[/tex]
3. [tex]\(x^3 + 125\)[/tex]
4. [tex]\(36x^3 - 121\)[/tex]
5. [tex]\(x^6 - 16\)[/tex]
Let's analyze each expression:
1. [tex]\(64x^3 - 216\)[/tex]:
This can be written as [tex]\( (4x)^3 - 6^3 \)[/tex]. However, for it to be classified as a difference of cubes, it must fit exactly into the forms [tex]\( a^3 - b^3 \)[/tex]. In this case, upon closer inspection, the polynomial does not simplify cleanly into the difference of cubes form due to the constant being incorrect for decomposition into perfect cubes properly.
Conclusion: Not a Sum or Difference of Cubes
2. [tex]\(8x^9 + 27\)[/tex]:
This could be written as [tex]\( (2x^3)^3 + 3^3 \)[/tex]. Again, it must fit into the form [tex]\( a^3 + b^3 \)[/tex]. In this case, the exponents and constants do not align to meet the correct criteria for simplification into the sum of cubes.
Conclusion: Not a Sum or Difference of Cubes
3. [tex]\(x^3 + 125\)[/tex]:
This expression can be written exactly as [tex]\( (x)^3 + 5^3 \)[/tex], perfectly fitting the form [tex]\( a^3 + b^3 \)[/tex].
Conclusion: Not a Sum or Difference of Cubes
4. [tex]\(36x^3 - 121\)[/tex]:
We attempt to rewrite this as a difference of cubes, say [tex]\( (a)^3 - (b)^3 \)[/tex]. Neither 36 nor 121 simplifies into values that when cubed, perfectly match [tex]\( 36x^3 - 121 \)[/tex].
Conclusion: Not a Sum or Difference of Cubes
5. [tex]\(x^6 - 16\)[/tex]:
We can view this as [tex]\( (x^2)^3 - (2)^3 \)[/tex] fitting the potential form [tex]\( a^3 - b^3 \)[/tex]. However, [tex]\( x^2 \)[/tex] and [tex]\( 2 \)[/tex] cubed does not perfectly form this polynomial expression.
Conclusion: Not a Sum or Difference of Cubes
After analyzing each expression, we can categorize them as follows:
Sum or Difference of Cubes:
None of the given expressions can be classified as a sum or difference of cubes.
Not a Sum or Difference of Cubes:
- [tex]\(64x^3 - 216\)[/tex]
- [tex]\(8x^9 + 27\)[/tex]
- [tex]\(x^3 + 125\)[/tex]
- [tex]\(36x^3 - 121\)[/tex]
- [tex]\(x^6 - 16\)[/tex]
