Answer: The required binomial that is a factor of f(x) is (x - 5).
Step-by-step explanation: We are given the binomials (x + 1), (x + 4), (x − 5), and (x − 2).
We are to select the one that is a factor of the following polynomial function :
[tex]f(x)=3x^3-12x^2-4x-55~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
Factor theorem : If the value of a function p(x) is zero at x = a, then (x - a) is a factor of p(x).
Now, substituting x = -1 in equation (i), we get
[tex]f(-1)\\\\=3(-1)^3-12(-1)^2-4(-1)-55\\\\=3\times (-1)-12\times1+4-55\\\\=-3-12-51\\\\=-66\neq 0.[/tex]
So, (x + 1) is NOT a factor f(x).
Substituting x = -4 in equation (i), we get
[tex]f(-4)\\\\=3(-4)^3-12(-4)^2-4(-4)-55\\\\=3\times (-64)-12\times16+16-55\\\\=-192-192-39\\\\=-384-66=-450\neq 0.[/tex]
So, (x + 4) is NOT a factor f(x).
Substituting x = 5 in equation (i), we get
[tex]f(5)\\\\=3(5)^3-12(5)^2-4(5)-55\\\\=3\times (125)-12\times25-20-55\\\\=375-300-75=0.[/tex]
So, (x - 5) is a factor f(x).
Substituting x = 2 in equation (i), we get
[tex]f(-4)\\\\=3(2)^3-12(2)^2-4(2)-55\\\\=3\times (8)-12\times4-8-55\\\\=24-48-63\\\\=-24-63=-87\neq 0.[/tex]
So, (x - 2) is NOT a factor f(x).
Thus, the required binomial that is a factor of f(x) is (x - 5).
