Question

Given the binomials (x + 1), (x + 4), (x − 5), and (x − 2), which one is a factor of f(x) = 3x3 − 12x2 − 4x − 55?

Answer 1

you have to do long division/synthetic division which I'm not going to try an type out on here.
(x - 5) (3x^2 + 3x + 11)

factor : (x - 5)

Answer 2

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 :

`f(x)=3x^3-12x^2-4x-55~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

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

`f(-1)\\\\=3(-1)^3-12(-1)^2-4(-1)-55\\\\=3* (-1)-12*1+4-55\\\\=-3-12-51\\\\=-66\\eq 0.`

So, (x + 1) is NOT a factor f(x).

Substituting x = -4 in equation (i), we get

`f(-4)\\\\=3(-4)^3-12(-4)^2-4(-4)-55\\\\=3* (-64)-12*16+16-55\\\\=-192-192-39\\\\=-384-66=-450\\eq 0.`

So, (x + 4) is NOT a factor f(x).

Substituting x = 5 in equation (i), we get

`f(5)\\\\=3(5)^3-12(5)^2-4(5)-55\\\\=3* (125)-12*25-20-55\\\\=375-300-75=0.`

So, (x - 5) is a factor f(x).

Substituting x = 2 in equation (i), we get

`f(-4)\\\\=3(2)^3-12(2)^2-4(2)-55\\\\=3* (8)-12*4-8-55\\\\=24-48-63\\\\=-24-63=-87\\eq 0.`

So, (x - 2) is NOT a factor f(x).

Thus, the required binomial that is a factor of f(x) is (x - 5).