How do you find the remainder without long division?

You can find the remainder without performing long division using the Remainder Theorem.

Remainder Theorem Explained

The Remainder Theorem offers a shortcut to finding the remainder when a polynomial, f(x), is divided by a linear divisor of the form x - a.

The core idea is:

• Set the divisor equal to zero: Solve the equation x - a = 0 for x. This gives you x = a.
• Substitute and evaluate: Substitute the value x = a into the polynomial f(x). The result, f(a), is the remainder. According to the provided reference, "another way to find the remainder is to set the x - a term equal to 0 and then solve for x. After this, you just plug it back in to find the remainder."

Example

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

1. Set the divisor equal to zero:
   x - 2 = 0
   x = 2

2. Substitute and evaluate:
   f(2) = (2)³ - 2(2)² + 5(2) - 7
   f(2) = 8 - 8 + 10 - 7
   f(2) = 3

Therefore, the remainder when x³ - 2x² + 5x - 7 is divided by x - 2 is 3.

Benefits

• Efficiency: It's often faster than long division, especially for simple divisors.
• Directness: It directly calculates the remainder without the intermediate steps of long division.