Polynomial long division might seem intimidating at first, but with a structured approach, it becomes manageable. This guide breaks down the process into easy-to-follow steps, equipping you with the skills to tackle any polynomial long division problem. We'll cover the method, provide examples, and offer tips for success.
Understanding Polynomial Long Division
Polynomial long division is a method used to divide a polynomial by another polynomial of a lower degree. It's similar to the long division you learned in elementary school, but instead of numbers, we're working with variables and exponents. The goal is to find the quotient and remainder after dividing the dividend (the polynomial being divided) by the divisor (the polynomial doing the dividing).
Key Terminology:
- Dividend: The polynomial being divided.
- Divisor: The polynomial doing the dividing.
- Quotient: The result of the division.
- Remainder: The amount left over after division.
Step-by-Step Process of Polynomial Long Division
Let's illustrate the process with an example: Divide (6x³ + 11x² - 17x - 10) by (2x + 5).
Step 1: Set up the problem:
Write the problem in the long division format:
2x + 5 | 6x³ + 11x² - 17x - 10
Step 2: Divide the leading terms:
Divide the leading term of the dividend (6x³) by the leading term of the divisor (2x):
6x³ / 2x = 3x²
Write this result (3x²) above the division bar, aligned with the x² term:
3x²
2x + 5 | 6x³ + 11x² - 17x - 10
Step 3: Multiply and subtract:
Multiply the quotient term (3x²) by the entire divisor (2x + 5):
3x²(2x + 5) = 6x³ + 15x²
Subtract this result from the dividend:
3x²
2x + 5 | 6x³ + 11x² - 17x - 10
- (6x³ + 15x²)
------------------
-4x² - 17x
Step 4: Bring down the next term:
Bring down the next term from the dividend (-17x):
3x²
2x + 5 | 6x³ + 11x² - 17x - 10
- (6x³ + 15x²)
------------------
-4x² - 17x
Step 5: Repeat steps 2-4:
Divide the leading term of the new dividend (-4x²) by the leading term of the divisor (2x):
-4x²/2x = -2x
Write this result (-2x) above the division bar:
3x² - 2x
2x + 5 | 6x³ + 11x² - 17x - 10
- (6x³ + 15x²)
------------------
-4x² - 17x
Multiply (-2x) by (2x + 5): -4x² - 10x
Subtract this from the current dividend:
3x² - 2x
2x + 5 | 6x³ + 11x² - 17x - 10
- (6x³ + 15x²)
------------------
-4x² - 17x
-(-4x² -10x)
-------------
-7x - 10
Bring down the next term (-10):
3x² - 2x
2x + 5 | 6x³ + 11x² - 17x - 10
- (6x³ + 15x²)
------------------
-4x² - 17x
-(-4x² -10x)
-------------
-7x - 10
Step 6: Final Division:
Divide -7x by 2x: -7/2
Multiply (-7/2) by (2x+5): -7x -35/2
Subtract:
3x² - 2x - 7/2
2x + 5 | 6x³ + 11x² - 17x - 10
- (6x³ + 15x²)
------------------
-4x² - 17x
-(-4x² -10x)
-------------
-7x - 10
-(-7x -35/2)
---------
15/2
The remainder is 15/2.
Step 7: Write the final answer:
The quotient is 3x² - 2x - 7/2, and the remainder is 15/2. The answer is often written as:
3x² - 2x - 7/2 + 15/2 / (2x+5)
Tips for Success with Polynomial Long Division
- Organize your work: Neatness is crucial. Keep your terms aligned and your calculations clear.
- Double-check your subtraction: Subtraction errors are common. Take your time and verify each step.
- Practice regularly: Like any math skill, practice makes perfect. Work through various problems to build your confidence.
By following these steps and practicing regularly, you'll master polynomial long division and confidently tackle more complex algebraic problems. Remember, the key is to break down the process into manageable steps and pay close attention to detail.
