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Sagot :
To determine which monomials are factors of [tex]\(-100 x^6 y\)[/tex], we need to carefully examine the given monomials and see if they divide [tex]\(-100 x^6 y\)[/tex] evenly. Let us analyze each option step-by-step:
### Option A: [tex]\(-8 x^2\)[/tex]
1. Identify the variables and their exponents:
- [tex]\(-100 x^6 y\)[/tex] has [tex]\(x\)[/tex] with an exponent of 6 and [tex]\(y\)[/tex] with an exponent of 1.
- [tex]\(-8 x^2\)[/tex] has [tex]\(x\)[/tex] with an exponent of 2 and no [tex]\(y\)[/tex] component.
2. Compare the exponents of [tex]\(x\)[/tex]:
- The exponent of [tex]\(x\)[/tex] in the candidate factor [tex]\(-8 x^2\)[/tex] is 2. This is less than the exponent of [tex]\(x\)[/tex] in [tex]\(-100 x^6 y\)[/tex] which is 6. This is good so far because 2 is a factor of 6.
3. Compare the constant coefficients:
- We need to check if [tex]\(-8\)[/tex] is a factor of [tex]\(-100\)[/tex].
- -100 divided by -8 does not result in an integer because [tex]\(100 / 8 = 12.5\)[/tex]; hence, -8 does not divide -100 evenly.
Since [tex]\(-8 x^2\)[/tex] does not divide [tex]\(-100 x^6 y\)[/tex] evenly due to the constant term, it is not a factor of [tex]\(-100 x^6 y\)[/tex].
### Option B: [tex]\(-4 x^3\)[/tex]
1. Identify the variables and their exponents:
- [tex]\(-100 x^6 y\)[/tex] has [tex]\(x\)[/tex] with an exponent of 6 and [tex]\(y\)[/tex] with an exponent of 1.
- [tex]\(-4 x^3\)[/tex] has [tex]\(x\)[/tex] with an exponent of 3 and no [tex]\(y\)[/tex] component.
2. Compare the exponents of [tex]\(x\)[/tex]:
- The exponent of [tex]\(x\)[/tex] in the candidate factor [tex]\(-4 x^3\)[/tex] is 3. This is less than the exponent of [tex]\(x\)[/tex] in [tex]\(-100 x^6 y\)[/tex] which is 6. This is acceptable because 3 is a factor of 6.
3. Compare the constant coefficients:
- We need to check if [tex]\(-4\)[/tex] is a factor of [tex]\(-100\)[/tex].
- -100 divided by -4 equals 25, which is an integer. So, [tex]\(-4\)[/tex] divides [tex]\(-100\)[/tex] evenly.
4. Check for the [tex]\(y\)[/tex] variable:
- The candidate factor [tex]\(-4 x^3\)[/tex] does not include the [tex]\(y\)[/tex] variable, while [tex]\(-100 x^6 y\)[/tex] does have [tex]\(y\)[/tex] with an exponent of 1.
- Since the factor [tex]\(-4 x^3\)[/tex] lacks the [tex]\(y\)[/tex] term that is present in [tex]\(-100 x^6 y\)[/tex], it does not divide it completely.
Considering the [tex]\(y\)[/tex] component, [tex]\(-4 x^3\)[/tex] is not a factor of [tex]\(-100 x^6 y\)[/tex].
### Conclusion
Based on the above analysis:
- Option A: [tex]\(-8 x^2\)[/tex] is not a factor of [tex]\(-100 x^6 y\)[/tex].
- Option B: [tex]\(-4 x^3\)[/tex] is not a factor of [tex]\(-100 x^6 y\)[/tex].
Therefore, the correct answer is: C) None of the above.
### Option A: [tex]\(-8 x^2\)[/tex]
1. Identify the variables and their exponents:
- [tex]\(-100 x^6 y\)[/tex] has [tex]\(x\)[/tex] with an exponent of 6 and [tex]\(y\)[/tex] with an exponent of 1.
- [tex]\(-8 x^2\)[/tex] has [tex]\(x\)[/tex] with an exponent of 2 and no [tex]\(y\)[/tex] component.
2. Compare the exponents of [tex]\(x\)[/tex]:
- The exponent of [tex]\(x\)[/tex] in the candidate factor [tex]\(-8 x^2\)[/tex] is 2. This is less than the exponent of [tex]\(x\)[/tex] in [tex]\(-100 x^6 y\)[/tex] which is 6. This is good so far because 2 is a factor of 6.
3. Compare the constant coefficients:
- We need to check if [tex]\(-8\)[/tex] is a factor of [tex]\(-100\)[/tex].
- -100 divided by -8 does not result in an integer because [tex]\(100 / 8 = 12.5\)[/tex]; hence, -8 does not divide -100 evenly.
Since [tex]\(-8 x^2\)[/tex] does not divide [tex]\(-100 x^6 y\)[/tex] evenly due to the constant term, it is not a factor of [tex]\(-100 x^6 y\)[/tex].
### Option B: [tex]\(-4 x^3\)[/tex]
1. Identify the variables and their exponents:
- [tex]\(-100 x^6 y\)[/tex] has [tex]\(x\)[/tex] with an exponent of 6 and [tex]\(y\)[/tex] with an exponent of 1.
- [tex]\(-4 x^3\)[/tex] has [tex]\(x\)[/tex] with an exponent of 3 and no [tex]\(y\)[/tex] component.
2. Compare the exponents of [tex]\(x\)[/tex]:
- The exponent of [tex]\(x\)[/tex] in the candidate factor [tex]\(-4 x^3\)[/tex] is 3. This is less than the exponent of [tex]\(x\)[/tex] in [tex]\(-100 x^6 y\)[/tex] which is 6. This is acceptable because 3 is a factor of 6.
3. Compare the constant coefficients:
- We need to check if [tex]\(-4\)[/tex] is a factor of [tex]\(-100\)[/tex].
- -100 divided by -4 equals 25, which is an integer. So, [tex]\(-4\)[/tex] divides [tex]\(-100\)[/tex] evenly.
4. Check for the [tex]\(y\)[/tex] variable:
- The candidate factor [tex]\(-4 x^3\)[/tex] does not include the [tex]\(y\)[/tex] variable, while [tex]\(-100 x^6 y\)[/tex] does have [tex]\(y\)[/tex] with an exponent of 1.
- Since the factor [tex]\(-4 x^3\)[/tex] lacks the [tex]\(y\)[/tex] term that is present in [tex]\(-100 x^6 y\)[/tex], it does not divide it completely.
Considering the [tex]\(y\)[/tex] component, [tex]\(-4 x^3\)[/tex] is not a factor of [tex]\(-100 x^6 y\)[/tex].
### Conclusion
Based on the above analysis:
- Option A: [tex]\(-8 x^2\)[/tex] is not a factor of [tex]\(-100 x^6 y\)[/tex].
- Option B: [tex]\(-4 x^3\)[/tex] is not a factor of [tex]\(-100 x^6 y\)[/tex].
Therefore, the correct answer is: C) None of the above.
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