分步说明
Divide the Highest Degree Term
Divide the highest degree term of the dividend by the highest degree term of the divisor
Multiply the Divisor by the Result
Multiply the entire divisor by the result from step 1
Subtract the Product from the Dividend
Subtract the product from step 2 from the dividend
Repeat the Process
Repeat the process with the new dividend and the divisor until the degree of the remainder is less than the degree of the divisor
Write the Final Quotient and Remainder
The final quotient is the sum of the results from each step, and the remainder is the final result from the subtraction step
Check Your Work
Use a calculator to check your work and ensure that the quotient and remainder are correct
Introduction to Polynomial Long Division
Polynomial long division is a method used to divide one polynomial by another. 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. This process is repeated until the degree of the remainder is less than the degree of the divisor.
Prerequisites
Before performing polynomial long division, it is essential to have a good understanding of polynomial operations, such as addition, subtraction, and multiplication.
The Formula
The formula for polynomial long division is: Dividend = Divisor * Quotient + Remainder
Worked Example
Let's divide the polynomial x^3 + 2x^2 - 7x - 12 by x + 3.
Step 1: Divide the Highest Degree Term
Divide the highest degree term of the dividend (x^3) by the highest degree term of the divisor (x). The result is x^2.
Step 2: Multiply the Divisor by the Result
Multiply the entire divisor (x + 3) by the result (x^2). This gives x^3 + 3x^2.
Step 3: Subtract the Product from the Dividend
Subtract the product (x^3 + 3x^2) from the dividend (x^3 + 2x^2 - 7x - 12). This gives -x^2 - 7x - 12.
Step 4: Repeat the Process
Repeat the process with the new dividend (-x^2 - 7x - 12) and the divisor (x + 3). Divide the highest degree term of the new dividend (-x^2) by the highest degree term of the divisor (x). The result is -x.
Step 5: Multiply the Divisor by the Result
Multiply the entire divisor (x + 3) by the result (-x). This gives -x^2 - 3x.
Step 6: Subtract the Product from the New Dividend
Subtract the product (-x^2 - 3x) from the new dividend (-x^2 - 7x - 12). This gives -4x - 12.
Step 7: Repeat the Process Again
Repeat the process with the new dividend (-4x - 12) and the divisor (x + 3). Divide the highest degree term of the new dividend (-4x) by the highest degree term of the divisor (x). The result is -4.
Step 8: Multiply the Divisor by the Result
Multiply the entire divisor (x + 3) by the result (-4). This gives -4x - 12.
Step 9: Subtract the Product from the New Dividend
Subtract the product (-4x - 12) from the new dividend (-4x - 12). This gives 0.
The final quotient is x^2 - x - 4, and the remainder is 0.
Common Mistakes to Avoid
When performing polynomial long division, it is essential to avoid common mistakes such as:
- Forgetting to subtract the product from the dividend
- Multiplying the divisor by the wrong result
- Not repeating the process until the degree of the remainder is less than the degree of the divisor
Using a Calculator for Convenience
While it is essential to understand how to perform polynomial long division manually, it can be convenient to use a calculator to check your work or to perform the calculation quickly. Most graphing calculators have a built-in polynomial long division function that can be used to divide polynomials quickly and accurately.