Answer :
To determine whether [tex]\((m - n)\)[/tex], [tex]\((n - p)\)[/tex], and [tex]\((p - m)\)[/tex] are factors of the function [tex]\( f(m, n, p) = m(n^2 - p^2) + \ln(p^2 - m^2) + p(m^2 - n^2) \)[/tex], we can use the factor theorem. The factor theorem states that if [tex]\( k \)[/tex] is a root of a polynomial [tex]\( f(x) \)[/tex], then [tex]\( (x - k) \)[/tex] is a factor of [tex]\( f(x) \)[/tex]. Here's how we can apply it systematically to each alleged factor:
### 1. Checking if [tex]\((m - n)\)[/tex] is a factor:
To determine if [tex]\((m - n)\)[/tex] is a factor, we substitute [tex]\( m = n \)[/tex] into the function and check if it equals zero.
[tex]\[ f(n, n, p) = n(n^2 - p^2) + \ln(p^2 - n^2) + p(n^2 - n^2) \][/tex]
Simplifying this expression:
[tex]\[ f(n, n, p) = n(n^2 - p^2) + \ln(p^2 - n^2) + p(0) \][/tex]
[tex]\[ f(n, n, p) = n(n^2 - p^2) + \ln(p^2 - n^2) \][/tex]
After substituting [tex]\( m = n \)[/tex], the result is:
[tex]\[ n(n^2 - p^2) + \ln(p^2 - n^2) \][/tex]
This expression is [tex]\( f(n, n, p) \)[/tex] which is not necessarily zero in general, but it confirms that [tex]\((m - n)\)[/tex] affects the structure significantly.
### 2. Checking if [tex]\((n - p)\)[/tex] is a factor:
To determine if [tex]\((n - p)\)[/tex] is a factor, we substitute [tex]\( n = p \)[/tex] into the function and check if it equals zero.
[tex]\[ f(m, p, p) = m(p^2 - p^2) + \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
Simplifying this expression:
[tex]\[ f(m, p, p) = m(0) + \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
[tex]\[ f(m, p, p) = \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
After substituting [tex]\( n = p \)[/tex], the result is:
[tex]\[ \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
This expression is [tex]\( f(m, p, p) \)[/tex] which is not necessarily zero in general, but it confirms that [tex]\((n - p)\)[/tex] affects the structure significantly.
### 3. Checking if [tex]\((p - m)\)[/tex] is a factor:
To determine if [tex]\((p - m)\)[/tex] is a factor, we substitute [tex]\( p = m \)[/tex] into the function and check if it equals zero.
[tex]\[ f(m, n, m) = m(n^2 - m^2) + \ln(m^2 - m^2) + m(m^2 - n^2) \][/tex]
Simplifying this expression:
[tex]\[ f(m, n, m) = m(n^2 - m^2) + \ln(0) + m(m^2 - n^2) \][/tex]
Noting that [tex]\(\ln(0)\)[/tex] is undefined (or tends to [tex]\(-\infty\)[/tex]), the expression is:
[tex]\[ f(m, n, m) = m(n^2 - m^2) + \ln(0) + m(m^2 - n^2) = \text{undefined} \][/tex]
Because [tex]\(\ln(0)\)[/tex] tends to [tex]\(-\infty\)[/tex], this suggests an indeterminate form, often represented as [tex]\( \text{zoo} \)[/tex] in symbolic computation, indicating that something breaks down regarding continuity or definition.
### Conclusion:
The function [tex]\( f(m, n, p) = m(n^2 - p^2) + \ln(p^2 - m^2) + p(m^2 - n^2) \)[/tex] provides us with specific results when examined for the factors [tex]\((m - n)\)[/tex], [tex]\((n - p)\)[/tex], and [tex]\((p - m)\)[/tex]:
- For [tex]\( (m - n) \)[/tex]: The expression simplifies to [tex]\( n(n^2 - p^2) + \ln(p^2 - n^2) \)[/tex]
- For [tex]\( (n - p) \)[/tex]: The expression simplifies to [tex]\( \ln(p^2 - m^2) + p(m^2 - p^2) \)[/tex]
- For [tex]\( (p - m) \)[/tex]: The expression results in an undefined form [tex]\( \text{zoo} \)[/tex]
These results confirm the structural impact of these terms, though they not always directly satisfy the classical factor theorem conditions strictly (i.e., being zero in well-posed cases).
### 1. Checking if [tex]\((m - n)\)[/tex] is a factor:
To determine if [tex]\((m - n)\)[/tex] is a factor, we substitute [tex]\( m = n \)[/tex] into the function and check if it equals zero.
[tex]\[ f(n, n, p) = n(n^2 - p^2) + \ln(p^2 - n^2) + p(n^2 - n^2) \][/tex]
Simplifying this expression:
[tex]\[ f(n, n, p) = n(n^2 - p^2) + \ln(p^2 - n^2) + p(0) \][/tex]
[tex]\[ f(n, n, p) = n(n^2 - p^2) + \ln(p^2 - n^2) \][/tex]
After substituting [tex]\( m = n \)[/tex], the result is:
[tex]\[ n(n^2 - p^2) + \ln(p^2 - n^2) \][/tex]
This expression is [tex]\( f(n, n, p) \)[/tex] which is not necessarily zero in general, but it confirms that [tex]\((m - n)\)[/tex] affects the structure significantly.
### 2. Checking if [tex]\((n - p)\)[/tex] is a factor:
To determine if [tex]\((n - p)\)[/tex] is a factor, we substitute [tex]\( n = p \)[/tex] into the function and check if it equals zero.
[tex]\[ f(m, p, p) = m(p^2 - p^2) + \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
Simplifying this expression:
[tex]\[ f(m, p, p) = m(0) + \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
[tex]\[ f(m, p, p) = \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
After substituting [tex]\( n = p \)[/tex], the result is:
[tex]\[ \ln(p^2 - m^2) + p(m^2 - p^2) \][/tex]
This expression is [tex]\( f(m, p, p) \)[/tex] which is not necessarily zero in general, but it confirms that [tex]\((n - p)\)[/tex] affects the structure significantly.
### 3. Checking if [tex]\((p - m)\)[/tex] is a factor:
To determine if [tex]\((p - m)\)[/tex] is a factor, we substitute [tex]\( p = m \)[/tex] into the function and check if it equals zero.
[tex]\[ f(m, n, m) = m(n^2 - m^2) + \ln(m^2 - m^2) + m(m^2 - n^2) \][/tex]
Simplifying this expression:
[tex]\[ f(m, n, m) = m(n^2 - m^2) + \ln(0) + m(m^2 - n^2) \][/tex]
Noting that [tex]\(\ln(0)\)[/tex] is undefined (or tends to [tex]\(-\infty\)[/tex]), the expression is:
[tex]\[ f(m, n, m) = m(n^2 - m^2) + \ln(0) + m(m^2 - n^2) = \text{undefined} \][/tex]
Because [tex]\(\ln(0)\)[/tex] tends to [tex]\(-\infty\)[/tex], this suggests an indeterminate form, often represented as [tex]\( \text{zoo} \)[/tex] in symbolic computation, indicating that something breaks down regarding continuity or definition.
### Conclusion:
The function [tex]\( f(m, n, p) = m(n^2 - p^2) + \ln(p^2 - m^2) + p(m^2 - n^2) \)[/tex] provides us with specific results when examined for the factors [tex]\((m - n)\)[/tex], [tex]\((n - p)\)[/tex], and [tex]\((p - m)\)[/tex]:
- For [tex]\( (m - n) \)[/tex]: The expression simplifies to [tex]\( n(n^2 - p^2) + \ln(p^2 - n^2) \)[/tex]
- For [tex]\( (n - p) \)[/tex]: The expression simplifies to [tex]\( \ln(p^2 - m^2) + p(m^2 - p^2) \)[/tex]
- For [tex]\( (p - m) \)[/tex]: The expression results in an undefined form [tex]\( \text{zoo} \)[/tex]
These results confirm the structural impact of these terms, though they not always directly satisfy the classical factor theorem conditions strictly (i.e., being zero in well-posed cases).
