Step-by-Step Instructions
Write the Dividend and Divisor
Write the dividend and divisor in the correct format, with the dividend on top and the divisor below.
Divide the Leading Term of the Dividend by the Leading Term of the Divisor
Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
Multiply the Entire Divisor by the Result
Multiply the entire divisor by the result from step 2 to get the product.
Subtract the Product from the Dividend
Subtract the product from the dividend to get the new dividend.
Repeat the Process
Repeat steps 2-4 with the new dividend until the degree of the remainder is less than the degree of the divisor.
Final Step
The final quotient is the sum of the terms obtained in each step, and the remainder is the final new dividend.
Introduction to Polynomial Long Division
Polynomial long division is a method used to divide one polynomial by another. It involves a series of steps that help us find the quotient and remainder of the division.
Prerequisites
Before we begin, make sure you have a good understanding of polynomial addition, subtraction, and multiplication. You should also be familiar with the concept of degrees of polynomials.
Step-by-Step Guide to Polynomial Long Division
To perform polynomial long division, follow these steps:
Steps to Perform Polynomial Long Division
The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, then multiplying the entire divisor by the result and subtracting it from the dividend.
Worked Example
Let's divide $x^3 + 2x^2 - 7x - 12$ by $x + 3$.
Step 1: Write the Dividend and Divisor
Write the dividend (the polynomial being divided) and the divisor (the polynomial by which we are dividing) in the following format:
_____________
x + 3 | x^3 + 2x^2 - 7x - 12
Step 2: Divide the Leading Term of the Dividend by the Leading Term of the Divisor
Divide the leading term of the dividend ($x^3$) by the leading term of the divisor ($x$).
x^2
x + 3 | x^3 + 2x^2 - 7x - 12
Step 3: Multiply the Entire Divisor by the Result
Multiply the entire divisor ($x + 3$) by the result ($x^2$).
x^2 * (x + 3) = x^3 + 3x^2
x + 3 | x^3 + 2x^2 - 7x - 12
-(x^3 + 3x^2)
Step 4: Subtract the Product from the Dividend
Subtract the product ($x^3 + 3x^2$) from the dividend ($x^3 + 2x^2 - 7x - 12$).
x + 3 | x^3 + 2x^2 - 7x - 12
-(x^3 + 3x^2)
_____________
-x^2 - 7x - 12
Step 5: Repeat the Process
Repeat steps 2-4 with the new dividend ($-x^2 - 7x - 12$).
-x^2
x + 3 | x^3 + 2x^2 - 7x - 12
-(x^3 + 3x^2)
_____________
-x^2 - 7x - 12
-(-x^2 - 3x)
_____________
-4x - 12
Step 6: Final Step
Divide the leading term of the new dividend ($-4x$) by the leading term of the divisor ($x$).
-4
x + 3 | x^3 + 2x^2 - 7x - 12
-(x^3 + 3x^2)
_____________
-x^2 - 7x - 12
-(-x^2 - 3x)
_____________
-4x - 12
-(-4x - 12)
_____________
0
The final quotient is $x^2 - x - 4$ with a remainder of $0$.
Common Mistakes to Avoid
- Forgetting to subtract the product from the dividend
- Not repeating the process until the degree of the remainder is less than the degree of the divisor
- Not writing the dividend and divisor in the correct format
When to Use a Calculator
While it's possible to perform polynomial long division by hand, it can be time-consuming and prone to errors. If you need to perform this calculation frequently or with large polynomials, consider using a calculator or computer algebra system for convenience.