1. Introduction
Division of polynomials means dividing one polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and possibly a remainder. This is similar to the long division of numbers but done using algebraic terms.
The division algorithm for polynomials states:
\( \, p(x) = d(x) \cdot q(x) + r(x) \, \)
where:
- \(p(x)\): dividend
- \(d(x)\): divisor
- \(q(x)\): quotient
- \(r(x)\): remainder
The degree of the remainder must be less than the degree of the divisor.
2. Steps of Polynomial Long Division
To divide polynomials, follow these steps:
- Arrange the terms of both dividend and divisor in decreasing powers of the variable.
- Divide the first term of the dividend by the first term of the divisor.
- Multiply the entire divisor by this result.
- Subtract the result from the dividend.
- Bring down the next term.
- Repeat until no terms are left or the degree of remainder is less than divisor.
2.1. Understanding with a Simple Pattern
Dividend ÷ Divisor → Quotient and Remainder
The idea is to reduce the dividend step by step by removing the highest-degree term each time.
3. Special Cases in Polynomial Division
Some divisions are easier or need special care.
3.1. Case 1: Dividing by a Monomial
If the divisor has only one term, divide each term of the polynomial by that term.
Example:
\( \dfrac{6x^3 - 9x^2 + 3x}{3x} = 2x^2 - 3x + 1 \)
3.2. Case 2: Dividing by a Binomial
Use the long division method carefully, aligning like terms correctly.
Example: Dividing by \(x - 2\).
3.3. Case 3: When the Remainder Appears Early
If the remaining term has lower degree than the divisor, it becomes the remainder and the division stops.
4. Worked Examples
Let’s divide polynomials using the long division method.
4.1. Example 1: Divide a Polynomial by a Monomial
Divide: \(6x^3 + 3x^2 - 9x \div 3x\)
- \(6x^3 ÷ 3x = 2x^2\)
- \(3x^2 ÷ 3x = x\)
- \(-9x ÷ 3x = -3\)
Quotient: \(2x^2 + x - 3\)
4.2. Example 2: Divide by a Binomial
Divide: \(x^3 - 3x^2 + 4x - 12\) by \(x - 3\)
Step 1: \(x^3 ÷ x = x^2\)
Multiply: \(x^2(x - 3) = x^2x - 3x^2 = x^3 - 3x^2\)
Subtract → new dividend: \(4x - 12\)
Step 2: \(4x ÷ x = 4\)
Multiply: \(4(x - 3) = 4x - 12\)
Subtract → remainder = 0
Quotient: \(x^2 + 4\)
4.3. Example 3: Division with Remainder
Divide: \(2x^3 + 3x^2 - x + 5\) by \(x + 2\)
Step 1: \(2x^3 ÷ x = 2x^2\)
Subtract to get new dividend: \( -x + 5 + ...\)
Continue division → final result:
Quotient: \(2x^2 - x + 1\)
Remainder: \(3\)
5. Division Algorithm for Polynomials
The division algorithm states:
\(p(x) = d(x)q(x) + r(x)\)
Where:
- \(p(x)\): dividend
- \(d(x)\): divisor
- \(q(x)\): quotient
- \(r(x)\): remainder (degree < degree of divisor)
6. Common Mistakes
- Not aligning terms by decreasing powers.
- Forgetting to subtract correctly during long division.
- Dividing only the first term instead of the entire expression.
- Incorrectly simplifying negative signs.
- Mixing up quotient and remainder.
7. Quick Practice
Try these questions:
- Divide \(4x^2 - 8x + 2\) by \(2x\).
- Divide \(x^3 + x^2 - x - 1\) by \(x + 1\).
- Divide \(3x^3 - 5x + 2\) by \(x - 2\).
- Verify the division algorithm for the above results.
8. Summary
- Polynomial division works like long division of numbers.
- Divide → Multiply → Subtract → Bring down → Repeat.
- Dividing by monomials is done term-by-term.
- Dividing by binomials requires careful long division.
- The division algorithm relates dividend, divisor, quotient, and remainder.