If $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ and if $(a_1-a_0)+(a_3-a_2)+\cdots =0$ , then one of the factors of  $f\left(x\right)$  is

If sum of the coefficients of even powers of 𝑥  and the constant term is equal to sum of the coefficients of odd powers of 𝑥 , then (𝑥 + 1)  is a factor.

$\therefore a_0+a_2+a_4+\cdots =a_1+a_3+a_5+\cdots$

$\Rightarrow (a_1-a_0)+(a_3-a_2)+(a_5-a_4)+\cdots =0$