Divisibility by $$(x-1)$$

Divisibility by $$(x-1)$$

## Formula

For a polynomial $P(x)$:

$$P(x)\text{ is divisible by }(x-1)\iff P(1)=0$$

For:

$$P(x)=(1+x)^n$$

the value at $x=-1$ is:

$$P(-1)=0$$

for:

$$n\ge1$$

Therefore:

$$(1+x)^n$$

is not itself divisible by $(x+1)$, but expressions such as:

$$(1+x)^n-1$$

are divisible by $x$.

## Usage

- Used in binomial divisibility problems and polynomial factor identification.