Question

state and prove remainder theorem​

Answer 1

Answer:

It states that the remainder of the division of the polynomial by a linear polynomial is equal to

Answer 2

Answer:

Definition:

The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x - a, the remainder of that division will be equivalent to f(a). In other words, if you want to evaluate the function f(x) for a given number, a, you can divide that function by x - a and your remainder will be equal to f(a).

Prove:

Consider that a polynomial P(x) after division results in D(x) . Q(x) + R where D is the divisor, Q is the quotient, and R is the remainder.

In this case, the polynomial is divided by x-k where k is a constant. This is the divisor,

so D(x) = x-k.

Substituting the divisor into the polynomial, we have:

P(x) = D(x) . Q(x) + R

P(x) = (x - k) . Q(x) + R

From here, you can easily see that P(k) = R:

P(k) = (k - k) . Q(x) + R

P(k) = R

hence proved