To determine which monomials are divisible by [tex]\(5x\)[/tex], we need to check if each monomial can be expressed as [tex]\(5x \cdot k(x)\)[/tex] for some polynomial [tex]\(k(x)\)[/tex]. This means:
1. The coefficient of the monomial must be divisible by 5.
2. The power of [tex]\(x\)[/tex] in the monomial must be at least 1 because [tex]\(5x\)[/tex] includes [tex]\(x\)[/tex] to the first power.
Let's analyze the given options:
### Option A: [tex]\(10 x^3\)[/tex]
1. Divisibility of the coefficient: The coefficient of [tex]\(10 x^3\)[/tex] is 10, which is divisible by 5.
2. Power of [tex]\(x\)[/tex]: The power of [tex]\(x\)[/tex] in [tex]\(10 x^3\)[/tex] is 3, which is greater than or equal to 1.
Thus, [tex]\(10 x^3\)[/tex] is divisible by [tex]\(5 x\)[/tex]:
[tex]\[
10 x^3 = 5 x \cdot (2 x^2)
\][/tex]
### Option B: [tex]\(5 x^2\)[/tex]
1. Divisibility of the coefficient: The coefficient of [tex]\(5 x^2\)[/tex] is 5, which is divisible by 5.
2. Power of [tex]\(x\)[/tex]: The power of [tex]\(x\)[/tex] in [tex]\(5 x^2\)[/tex] is 2, which is greater than or equal to 1.
Thus, [tex]\(5 x^2\)[/tex] is divisible by [tex]\(5 x\)[/tex]:
[tex]\[
5 x^2 = 5 x \cdot x
\][/tex]
### Option C: None of the above
Since both options A and B are divisible by [tex]\(5 x\)[/tex], option C is incorrect.
### Conclusion
The monomials that are divisible by [tex]\(5 x\)[/tex] are:
- Option A: [tex]\(10 x^3\)[/tex]
- Option B: [tex]\(5 x^2\)[/tex]
Thus, the correct answers are:
[tex]\( \boxed{10 x^3} \)[/tex] and [tex]\( \boxed{5 x^2} \)[/tex].
