Question

How do you determine whether x+2 is a factor of the polynomial `2x^3+2x^2-x-6`?

Answer 1

You divide using polynomial long division or synthetic division and if the remainder is 0 then it is a factor.

Explanation:

`2x^3+2x^2-x-6 divide x+2`

this division has a remainder of -12 so it is not a factor.

Answer 2

Just check if `x=-2` is a root of polynomial. See below

Explanation:

We know that if a polynomial `P(x)` has a root for `x=a`, then `P(a)=0` and `x-a` divides to `P(x)` (this is the same that `x-a` is a factor of `P(x)`

Lets see.

`P(-2)=2·(-2)^3+2·(-2)^2-(-2)-6=-16+8+2-6=-12!=0`

Then `x+2` is not a factor of `P(x)`

Answer 3

`x+2` not a factor of `f(x)`

Explanation:

Let `f(x)=2x^3+2x^2-x-6`

We can use the Polynomial Remainder Theorem, which states when a polynomial, `f(x)` is divided by `x-c`, the remainder is `f(c)`.

We are dividing `f(x)` by `x+2`, so `c=-2`. Now, let's input this into `f(x)`. We get

`2(-2)^3+2(-2)^2-(-2)-6`

`=>2(-8)+2(4)+2-6`

`=>-16+8+2-6`

`=>color(blue)(-12)`

We have a remainder, which means `x+2` is not a factor of `f(x)`.

If `f(c)` simplified to `0`, we would have no remainder, and `x+2` would be a factor, but since we have a remainder, `x+2` is not a factor.

Hope this helps!