Prove that a polynomial `p(x)` is divisible by `(x-1)` if the sum of coefficients of `p(x)` is zero.

Expert Answers
degeneratecircle eNotes educator| Certified Educator

Let the polynomial be


We see that


According to the assumption that the coefficients add to zero, we have `p(1)=0.`

If `p(1)=0,` then by the Factor Theorem, `(x-1)` divides `p(x).`

oldnick | Student

first if `(x-1) |p(x)`   is that: `p(x)=(x-1)q(x)` ,so that:


and  `p(x)=sum_(k=0)^n a_k x^k`  (1)


   `p(1)=sum_(k=0)^na_k=0`   (2)


On the other side, if `p(x) ` verifies (2) it means `p(1)=0` 

So we know from polynomial rules:

`p(x)-p(1)=q(x)(x-1)`   (3)

Being: `p(1)=0` :

`p(x)=q(x)(x-1)`  that means  `(x-1)|p(x)` 

so we can affirm:

A polynom `p(x)=sum_(k=0)^n a_kx^k` 

verifies:   `sum_(k=0)^na_k=0`  if only if,  `(x-1)|p(x)`