Question

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

Accepted Answer

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)

Answer

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.

Answer

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.

Math

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

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