Dividing polynomials can be approached using methods similar to long division with numbers, and synthetic division offers a more streamlined approach, especially for linear divisors. Here's a breakdown of both methods:
Long Division of Polynomials
Long division for polynomials mirrors the arithmetic long division.
Steps:
- Arrange: Write the dividend (the polynomial to be divided) and the divisor (what you are dividing by) in descending order of their powers. If any terms are missing in the dividend, insert terms with a coefficient of 0 as placeholders.
- Divide: Divide the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient.
- Multiply: Multiply the entire divisor by the first term of the quotient, and write the resulting polynomial below the dividend.
- Subtract: Subtract the result from the dividend, being careful to change all signs of the terms you are subtracting.
- Bring Down: Bring down the next term from the original dividend.
- Repeat: Repeat the process from step 2 until all the terms in the dividend have been brought down, or until the degree of the remainder is less than the degree of the divisor.
Example: Divide (6x³ - 19x² + 16x - 4) by (2x - 3)
| 3x² -5x +1 | |
|---|---|
| 2x-3 | 6x³ - 19x² + 16x - 4 |
| -(6x³ - 9x²) | |
| -10x² + 16x | |
| -(-10x² +15x) | |
| x -4 | |
| -(x -3/2) | |
| -5/2 |
- Result: The quotient is 3x² - 5x + 1, and the remainder is -5/2. The answer is written as 3x² - 5x + 1 - 5/(2(2x-3)).
Synthetic Division of Polynomials
Synthetic division provides a simpler method for dividing a polynomial by a linear divisor of the form x - k.
Steps: (Referencing the provided material)
- Identify k: Write k for the divisor, derived from x - k. If the divisor is x + k, then k will be negative. The divisor must be in the form of (x-k) or the method will not work.
- List Coefficients: Write down the coefficients of the dividend. Ensure you include zeros as placeholders for any missing terms.
- Bring Down: Bring down the leading coefficient of the dividend below the line.
- Multiply: Multiply the leading coefficient by k and write the result under the next coefficient of the dividend.
- Add: Add the terms in the second column and write the result below the line.
- Repeat: Multiply the new result by k and write it under the next coefficient, repeating the steps 4 and 5 for all remaining terms.
Example: Divide (x³ - 7x - 6) by (x - 3)
-
k = 3
-
Coefficients: 1, 0, -7, -6
3 1 0 -7 -6 3 9 6 1 3 2 0
- Result: The quotient is x² + 3x + 2, and the remainder is 0.
Key Points:
- Synthetic division is faster but can only be used if the divisor is linear with a leading coefficient of 1, that is of the form x-k. For more complex divisors, long division is necessary.
- Always arrange the terms in descending order of their exponents in both the dividend and the divisor.
- Pay close attention to signs during subtraction in long division.
