Prove that when a polynomial function p(x) is divided by a first degree polynomial ax + b the remainder is p(-b/a)

r = p(-b/a)

Step-by-step explanation:

Let p(x) = q(x)(ax + b) + r(x) where q(x) is the quotient when p(x) is divided by ax + b and r(x) is the remainder.

Since ax + b is a first degree polynomial, r(x) is one power less than ax + b is is just a constant, r.

So, p(x) = q(x)(ax + b) + r

Now, p(x) = r when q(x)(ax + b) = 0

since q(x) ≠ 0, ax + b = 0 ⇒ ax = -b ⇒ x = -b/a

⇒ p(x) = r when x = -b/a

So, r = p(-b/a)

So, the remainder when a polynomial function p(x) is divided by a first degree polynomial ax + b is p(-b/a)