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?

(x + 1)

(x + 4)

(x - 5)

(x - 2)

Answer 1

`(x-5)`

Explanation:

The Factor Theorem tells us that substituting
`x=a` into a polynomial
`p(x)` when divided by a linear factor
`(x-a)` will give us a value of 0 IF
`(x-a)` is a factor of the polynomial
`p(x)`

So, checking each of the 4 binomials, and getting a 0 as answer will tell us which one is a factor of the function given.

Putting
`x=-1` for
`(x+1)` into the function:

`3(-1)^(3)-12(-1)^2-4(-1)-55\\=-66`

NOT a factor.

Putting
`x=-4` for
`(x+4)` into the function:

`3(-4)^(3)-12(-4)^2-4(-4)-55\\=-423`

NOT a factor.

Putting
`x=5` for
`(x-5)` into the function:

`3(5)^(3)-12(5)^2-4(5)-55\\=0`

IS a factor.

Putting
`x=2` for
`(x-2)` into the function:

`3(2)^(3)-12(2)^2-4(2)-55\\=-87`

NOT a factor.

So we can see that only
`(x-5)` is a factor.

Answer 2

For this case we have the following polynomial:

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

The first thing we must do is factor the polynomial.

We have then:

`f (x) = (x-5) (3x ^ 2 + 3x + 11)`

Therefore, we observe that the common factor factor of the polynomial is:

`(x-5)`

Answer:

The common factor factor of the given function is:

`(x-5)`