If P(x) is a polynomial, then P(r) = 0 if and only if x - r is a ________(what) of P(x)?

Expert Answers
hala718 eNotes educator| Certified Educator

P(x) is a polynomial.

Assume that P(x) is a combination of two factors

then P(x)= f(x) *g(x)

now we know that P(r)= 0, then r is a root to p(x)

then, P(x) = (x-r) g(x)

then (x-r)= p(x)/g(x) where g(x) is a factor

then (x-r) is a factor of P(x)

giorgiana1976 | Student

If P(r)=0, that means that r is a root of P(x), because w eknow that if we substitute the root of an expression into it, the expression will be cancelled.

We'll write the rule of division with reminder and we'll have:

P(x)=(x-r)*Q(x) + R(x)

We observe that if we substitute x by r, we'll get:

P(r)=(r-r)*Q(r) + R(r)

P(r)=0*Q(r) + R(r)

P(r)=R(r)

But P(r)=0, so R(r)=0.

From here, we conclude that P(x) is divisible by (x-r) and (x-r) is P(x) linear factor.

We know that an expression could be written as a product of linear factors.

neela | Student

If x-r should be a factor of p(x), then P(r) = 0

Proof:

If x-r i s factor of xero, then

P(x) / (x-r) = Q(x) and  no remainder. Therefore

P(x) = (x-r)Q(x). Put x =r on both sides an d we get:

P(r) = (r-r)Q(r). Or

P(r) = 0*Q(R). Or

P(r) = 0. When x-r is a factor.