Polynomial division, similar to long division with numbers, involves dividing a polynomial (the dividend) by another polynomial (the divisor). It can be approached through long division or synthetic division (when the divisor is a linear factor). Here’s a breakdown of the long division method, drawing from the provided video reference.
Polynomial Long Division: A Step-by-Step Guide
Polynomial long division method is a technique to divide a polynomial by another polynomial, very similar to numerical long division. Here’s how it works, with an example inspired by the reference:
Steps:
- Set up the Problem: Write the dividend inside the long division symbol and the divisor outside. Make sure to include placeholders for any missing terms in both the dividend and the divisor (e.g.
0xor0x^2). This helps keep things aligned. - Divide Leading Terms: Divide the leading term of the dividend by the leading term of the divisor. This result becomes the first term of the quotient (the answer).
- Multiply: Multiply the term you just found in the quotient by the entire divisor.
- Subtract: Subtract the result of the multiplication from the corresponding terms of the dividend. Remember to change the signs when subtracting.
- Bring Down: Bring down the next term from the dividend.
- Repeat: Repeat steps 2 through 5 using the new polynomial you obtained after the subtraction until there are no more terms to bring down, or until the degree of the remaining polynomial is lower than the degree of the divisor.
- Remainder: The remaining polynomial after the last subtraction is the remainder. If it is zero then the division is exact.
Example Based on Reference:
Let's see a simplified example inspired from the reference material. Suppose we are dividing 6x^4 + 4x^3 + 5x^2 + 11x + 2 by 2x^2 + 2x + 1
| Step | Action | Calculation | Result (Quotion and Remainder) |
|---|---|---|---|
| 1 | Set up | 2x^2+2x+1 |
6x^4+4x^3+5x^2+11x+2 |
| 2 | Divide leading terms | 6x^4 / 2x^2 = 3x^2 |
Quotient: 3x^2 |
| 3 | Multiply quotient term by divisor | 3x^2(2x^2 + 2x + 1) = 6x^4 + 6x^3 + 3x^2 |
6x^4+4x^3+5x^2+11x+2 - (6x^4 + 6x^3 + 3x^2) |
| 4 | Subtract | (6x^4+4x^3+5x^2)-(6x^4 + 6x^3 + 3x^2)= -2x^3 +2x^2 |
-2x^3 +2x^2 + 11x + 2 |
| 5 | Bring down | Bring down + 11x and + 2 |
-2x^3 +2x^2 + 11x + 2 |
| 6 | Divide Leading term of new polynomial by divisor | -2x^3/2x^2 = -x |
Quotient: 3x^2 -x |
| 7 | Multiply quotient term by divisor | -x (2x^2 + 2x + 1)= -2x^3 -2x^2 -x |
-2x^3 +2x^2 + 11x + 2 - (-2x^3 -2x^2 -x) |
| 8 | Subtract | (-2x^3 +2x^2 + 11x) - (-2x^3 -2x^2 -x) = 4x^2 +12x |
4x^2 +12x+2 |
| 9 | Bring down | Bring down + 2 |
4x^2 +12x + 2 |
| 10 | Divide Leading term of new polynomial by divisor | 4x^2/2x^2 = 2 |
Quotient: 3x^2 -x +2 |
| 11 | Multiply quotient term by divisor | 2(2x^2 + 2x + 1) = 4x^2 + 4x + 2 |
4x^2 +12x + 2 - (4x^2 + 4x + 2) |
| 12 | Subtract | (4x^2 +12x + 2) - (4x^2 + 4x + 2) = 8x |
8x |
Therefore, the result of the division is quotient 3x^2 - x + 2 and the remainder is 8x. So (6x^4 + 4x^3 + 5x^2 + 11x + 2) / (2x^2 + 2x + 1) = 3x^2 - x + 2 + 8x/(2x^2 + 2x + 1)
Key Considerations:
- Placeholders: Always include placeholders (terms with coefficient 0) for missing powers of x in both the dividend and divisor. This keeps columns aligned.
- Sign Changes: When subtracting, remember to change the sign of each term in the polynomial being subtracted.
Practical Tips:
- Organization: Use graph paper or lined paper to help keep the columns neatly aligned, reducing errors.
- Check Your Work: After division, you can check your answer by multiplying the quotient by the divisor and adding the remainder. The result should equal the dividend.
Polynomial division is a foundational concept in algebra, allowing manipulation of polynomial equations and expressions.
