How Do You Find the Remainder in the Remainder Theorem?

The remainder in the Remainder Theorem is found by evaluating the polynomial at the value that makes the divisor equal to zero.

Understanding the Remainder Theorem

The Remainder Theorem provides a shortcut for finding the remainder when a polynomial, p(x), is divided by a linear divisor of the form (x - a). Instead of performing long division, you can directly calculate the remainder.

The Process

1. Identify the divisor: This is usually in the form (x - a).

2. Solve for 'a': Set the divisor equal to zero and solve for x. This gives you the value 'a'. For example, if the divisor is (x - 3), then a = 3.

3. Evaluate p(a): Substitute the value of 'a' into the polynomial p(x). The result, p(a), is the remainder.

Formula

Mathematically, the Remainder Theorem can be expressed as:

If p(x) is divided by (x - a), then the remainder is p(a).

Example

Let's say you want to find the remainder when the polynomial p(x) = x³ - 2x² + 5x - 7 is divided by (x - 2).

1. Divisor: (x - 2)

2. Solve for 'a': x - 2 = 0 => x = 2. So, a = 2.

3. Evaluate p(a): p(2) = (2)³ - 2(2)² + 5(2) - 7 = 8 - 8 + 10 - 7 = 3

Therefore, the remainder is 3.

Why it Works

The Remainder Theorem is based on the division algorithm for polynomials, which states:

Dividend = (Divisor × Quotient) + Remainder

Using this, we can write:

p(x) = (x - a) * q(x) + r

where:

• p(x) is the polynomial
• (x - a) is the divisor
• q(x) is the quotient
• r is the remainder (a constant, since we're dividing by a linear term)

If we substitute x = a into the equation, we get:

p(a) = (a - a) q(a) + r
p(a) = 0 q(a) + r
p(a) = r

This shows that p(a) is indeed the remainder.

In summary:

To find the remainder using the Remainder Theorem, simply evaluate the polynomial at the value that makes the linear divisor equal to zero. This evaluation directly yields the remainder of the division.