To find the number of imaginary (complex) solutions for each of the given polynomial equations, we will analyze each polynomial individually and determine the characteristics of their roots.
### a) Polynomial: [tex]\( x^4 - x^3 - 13x^2 + x + 12 \)[/tex]
This is a fourth-degree polynomial (quartic polynomial). Generally, a polynomial of degree [tex]\( n \)[/tex] has [tex]\( n \)[/tex] roots (real or complex). However, when we solve this specific polynomial, we find that all of its roots are real numbers. Thus, the number of imaginary roots is:
[tex]\[ \boxed{0} \][/tex]
### b) Polynomial: [tex]\( x^5 + x^4 - 15x^3 - 25x^2 + 14x + 24 \)[/tex]
This is a fifth-degree polynomial. Similar to the previous case, we count the total number of roots and their nature. Upon solving, it is determined that all roots of this polynomial are real. Hence, the number of imaginary roots is:
[tex]\[ \boxed{0} \][/tex]
### c) Polynomial: [tex]\( x^3 - 13x - 12 \)[/tex]
This is a third-degree polynomial (cubic polynomial). Again, we analyze the roots and find that all roots are real. Therefore, the number of imaginary roots is:
[tex]\[ \boxed{0} \][/tex]
### d) Polynomial: [tex]\( x^2 - 3x - 4 \)[/tex]
This is a second-degree polynomial (quadratic polynomial). By analyzing the roots, we see that they are both real. Consequently, the number of imaginary roots is:
[tex]\[ \boxed{0} \][/tex]
### e) Polynomial: [tex]\( -x + 4 \)[/tex]
This is a first-degree polynomial (linear polynomial). It has only one root, which is real. Thus, the number of imaginary roots is:
[tex]\[ \boxed{0} \][/tex]
### Summary
For each polynomial, we determined that none of the roots are imaginary. Therefore, the number of imaginary (complex) solutions separated by commas for each polynomial are:
[tex]\[ \boxed{0, 0, 0, 0, 0} \][/tex]
These steps and conclusions agree with the given answers.
