To determine which of the following is a factor of the given expression [tex]\(500x^3 + 108y^{18}\)[/tex], let's perform a detailed factorization of the expression.
Given:
[tex]\[ 500x^3 + 108y^{18} \][/tex]
1. First, we look for any common factors between the terms. Notice that both terms can be factored by 4:
[tex]\[ 500x^3 + 108y^{18} = 4 \cdot 125x^3 + 4 \cdot 27y^{18} \][/tex]
[tex]\[ = 4 (125x^3 + 27y^{18}) \][/tex]
2. Next, we factorize [tex]\(125x^3 + 27y^{18}\)[/tex]. By recognizing that these are both perfect cubes, we can use the sum of cubes factorization formula:
The sum of cubes formula is [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex].
For the expression [tex]\(125x^3 + 27y^{18}\)[/tex]:
- Let [tex]\(a = 5x\)[/tex]
- Let [tex]\(b = 3y^6\)[/tex]
Then:
[tex]\[ 125x^3 = (5x)^3 \][/tex]
[tex]\[ 27y^{18} = (3y^6)^3 \][/tex]
So we can rewrite the expression as:
[tex]\[ 125x^3 + 27y^{18} = (5x)^3 + (3y^6)^3 \][/tex]
Applying the sum of cubes formula:
[tex]\[ = (5x + 3y^6)\left((5x)^2 - (5x)(3y^6) + (3y^6)^2\right) \][/tex]
[tex]\[ = (5x + 3y^6)(25x^2 - 15xy^6 + 9y^{12}) \][/tex]
Putting everything together, we have:
[tex]\[ 500x^3 + 108y^{18} = 4 (5x + 3y^6)(25x^2 - 15xy^6 + 9y^{12}) \][/tex]
Thus, the complete factorization is:
[tex]\[ 4(5x + 3y^6)(25x^2 - 15xy^6 + 9y^{12}) \][/tex]
From this factorization, we can see the factors of [tex]\(500x^3 + 108y^{18}\)[/tex]. The following are the factors:
1. [tex]\(4\)[/tex]
2. [tex]\(5x + 3y^6\)[/tex]
3. [tex]\(25x^2 - 15xy^6 + 9y^{12}\)[/tex]
Therefore, any of the trivial factors could potentially be given as options. One of the primary non-trivial factors from the expression is [tex]\((5x + 3y^6)\)[/tex], which is indeed a factor of [tex]\(500x^3 + 108y^{18}\)[/tex].
