The long division method for polynomials, often taught in Class 9, involves dividing one polynomial by another, similar to dividing numbers. Here’s a step-by-step guide, incorporating information from the provided reference:
Understanding Polynomial Long Division
Polynomial long division is a technique to divide a polynomial (the dividend) by another polynomial (the divisor), resulting in a quotient and a remainder. It's a core concept in algebra.
Steps for Polynomial Long Division
- Arrange the Polynomials:
- Sort the terms: Arrange the terms of both the dividend and the divisor in descending order based on the degree of the variable (the exponents). For example, instead of
2x + 3x² + 1, write3x² + 2x + 1. - Fill in missing terms: If a term is missing (e.g., there's no x term), use a zero coefficient to hold its place. For instance, if you have
x³ + 1you would writex³ + 0x² + 0x + 1.
- Sort the terms: Arrange the terms of both the dividend and the divisor in descending order based on the degree of the variable (the exponents). For example, instead of
- Set Up the Division:
- Write the divisor to the left and the dividend under the division bar.
- Divide the Leading Terms:
- Find the quotient's first term: Divide the first term of the dividend by the first term of the divisor. This result will be the first term of your quotient.
- Multiply the Quotient by the Divisor:
- Multiply the new term of the quotient by the entire divisor. Write the result below the dividend, aligning terms with the same degree.
- Subtract and Bring Down:
- Subtract the product you calculated in step 4 from the dividend. Change the signs of each term being subtracted to facilitate the process.
- Bring down the next term of the original dividend to the remainder from the subtraction step.
- Repeat the Process:
- Use the new remainder as your new dividend. Repeat steps 3-5 until the degree of the remainder is less than the degree of the divisor.
- Write the Result:
- The polynomial above the bar is the quotient, and the remaining polynomial is the remainder.
- The result is written as: Dividend = (Divisor × Quotient) + Remainder, or Quotient + (Remainder / Divisor).
Example
Let's divide (x² + 5x + 6) by (x + 2):
| Step | Action | Explanation |
|---|---|---|
| 1 | Divide the leading terms: x² / x = x |
This x is the first term of the quotient |
| 2 | Multiply the quotient term by divisor: x * (x + 2) = x² + 2x |
Write this below the dividend. |
| 3 | Subtract the result from the dividend: (x² + 5x + 6) - (x² + 2x) = 3x + 6 |
Bring down the next term (6). |
| 4 | Divide the new leading terms: 3x / x = 3 |
Write 3 in the quotient as the second term |
| 5 | Multiply the new term by the divisor: 3 * (x + 2) = 3x + 6 |
Write this below the 3x + 6 |
| 6 | Subtract from the remainder (3x + 6) - (3x + 6) = 0 |
The remainder is 0 |
Therefore, the quotient is x + 3, and the remainder is 0.
Practical Insights
- Checking Your Work: Always verify your work by confirming that
(Divisor * Quotient) + Remainderequals the Dividend. - Care with Signs: Be extremely careful with signs during the subtraction phase; this is where most mistakes happen.
- Practice Makes Perfect: Like all math skills, mastering long division requires consistent practice.
By diligently following these steps, you can accurately perform polynomial long division, a valuable skill for algebra in Class 9 and beyond.
