What is the Remainder theorem for polynomials?

The Remainder theorem states that the remainder of a polynomial division can be found by substituting the value of x into the polynomial.

When dividing a polynomial f(x) by a linear factor (x-a), the Remainder theorem states that the remainder R can be found by substituting a into f(x). Mathematically, this can be written as:

f(x) = (x-a)q(x) + R

where q(x) is the quotient of the division. To find R, we substitute a into f(x) and simplify:

f(a) = (a-a)q(a) + R
f(a) = 0q(a) + R
R = f(a)

For example, let's find the remainder when f(x) = x^3 - 2x^2 + 3x - 4 is divided by (x-2). Using the Remainder theorem, we substitute 2 into f(x) to find the remainder:

f(2) = 2^3 - 2(2)^2 + 3(2) - 4
f(2) = 8 - 8 + 6 - 4
f(2) = 2

Therefore, the remainder when f(x) is divided by (x-2) is 2.

The Remainder theorem can also be used to determine whether a given value is a root of a polynomial. If the remainder is 0, then the value is a root. For example, if the remainder when f(x) is divided by (x-3) is 0, then 3 is a root of f(x).