Answer :
The polynomial [tex]\(x^3 + 64\)[/tex] is an example of a sum of cubes. In mathematics, a sum of cubes refers to an expression of the form [tex]\(a^3 + b^3\)[/tex].
Here's how we identify that [tex]\(x^3 + 64\)[/tex] is a sum of cubes:
1. Recognize the Structure:
- We have the polynomial [tex]\(x^3 + 64\)[/tex].
- Notice that 64 is a perfect cube, since [tex]\(4 \times 4 \times 4 = 64\)[/tex], which can also be written as [tex]\(4^3\)[/tex].
2. Express Each Term as a Cube:
- The term [tex]\(x^3\)[/tex] is already in the form of a cube, [tex]\((x)^3\)[/tex].
- The number 64 can be expressed as [tex]\((4)^3\)[/tex].
3. Rewrite the Polynomial Using These Cubes:
- Using the identified cubes, we can rewrite the polynomial as [tex]\((x)^3 + (4)^3\)[/tex].
So, the polynomial [tex]\(x^3 + 64\)[/tex] can be expressed as the sum of cubes [tex]\((x)^3 + (4)^3\)[/tex].
Here's how we identify that [tex]\(x^3 + 64\)[/tex] is a sum of cubes:
1. Recognize the Structure:
- We have the polynomial [tex]\(x^3 + 64\)[/tex].
- Notice that 64 is a perfect cube, since [tex]\(4 \times 4 \times 4 = 64\)[/tex], which can also be written as [tex]\(4^3\)[/tex].
2. Express Each Term as a Cube:
- The term [tex]\(x^3\)[/tex] is already in the form of a cube, [tex]\((x)^3\)[/tex].
- The number 64 can be expressed as [tex]\((4)^3\)[/tex].
3. Rewrite the Polynomial Using These Cubes:
- Using the identified cubes, we can rewrite the polynomial as [tex]\((x)^3 + (4)^3\)[/tex].
So, the polynomial [tex]\(x^3 + 64\)[/tex] can be expressed as the sum of cubes [tex]\((x)^3 + (4)^3\)[/tex].
