We use the Factor and Remainder theorem to solve this problem. When a polynomial function P(x) is divided by a monomial, Q(x), you equate the monomial to zero to solve x. Then, you substitute this value to P(x). If the function equals zero, there is no remainder. If the function equals to an integer, that integer is the remainder. Let's perform these procedures.
[tex]P(x) = x^{4}+ x^{3}-13 x^{2} -25x-12 [/tex]
[tex]Q(x)=x-4=0[/tex]
[tex]x=4[/tex]
Substituting x to P(x),
[tex]P(4) = 4^{4}+ 4^{3}-13 (4)^{2} -25(4)-12=0[/tex]
This means that there is no remainder.
