To find the remainder of a polynomial division, you can use several methods, the most common being polynomial long division and the Remainder Theorem.
Methods to Find the Remainder
Here's a breakdown of how to find the remainder:
1. Polynomial Long Division
This method is similar to long division with numbers. It involves dividing the polynomial by the divisor step-by-step until you reach a remainder that has a lower degree than the divisor.
- Process: Perform long division as you would with numbers.
- Result: The final expression left over after the division is the remainder.
2. Remainder Theorem
The Remainder Theorem provides a shortcut for finding the remainder when dividing by a linear divisor of the form x – c.
- Theorem: If the polynomial P(x) is divided by x – c, then the remainder is the value P(c). This means you substitute c into the polynomial P(x) to find the remainder.
Example Using the Remainder Theorem
Let's say we want to find the remainder when P(x) = x3 + 2x2 - 5x + 1 is divided by x - 2.
- Identify c: In this case, x - c = x - 2, so c = 2.
- Evaluate P(c): Substitute c = 2 into P(x).
- P(2) = (2)3 + 2(2)2 - 5(2) + 1
- P(2) = 8 + 8 - 10 + 1
- P(2) = 7
Therefore, the remainder when x3 + 2x2 - 5x + 1 is divided by x - 2 is 7.
Choosing the Right Method
- Use the Remainder Theorem when you're dividing by a linear expression of the form x - c. It's quicker and more efficient.
- Use Polynomial Long Division when dividing by a polynomial of a higher degree or when you need to find both the quotient and the remainder.
