Unit: Factorization Of Expressions
Chapter: Division of Algebraic Expressions
Reference: – Introduction to Division of Algebraic Expressions, Dividing a Monomial by a Monomial, Dividing a Polynomial by a Monomial, Dividing a Polynomial by a Polynomial, Using Factorization for Division, Cancellation of Common Factors, Division Algorithm (Quotient and Remainder), Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- How to Divide Monomials
- How to Divide a Polynomial by a Monomial
- How to Divide Polynomials Using Factorization
- How to Cancel Common Factors
Introduction to Division of Algebraic Expressions
Definition
Division of algebraic expressions is the process of dividing one algebraic expression by another. Just like with numbers, division is the inverse of multiplication. For example, if 3x × (x + 2) = 3x² + 6x, then (3x² + 6x) ÷ 3x = x + 2.
When we divide algebraic expressions, we essentially ask:
"What expression, when multiplied by the divisor, gives the dividend?"
Division can often be simplified by factoring both the dividend and divisor and cancelling common factors.
Importance of Division of Algebraic Expressions
- Simplifies algebraic fractions
- Essential for solving rational equations
- Used in polynomial long division (higher grades)
- Helps understand the relationship between multiplication and division
- Foundation for calculus (derivatives of rational functions)
Example
15x³ ÷ 3x² = (15/3) × (x³/x²) = 5x
Subtopics
1. Dividing a Monomial by a Monomial
A monomial is an expression with one term (like 6x², 10y³, -4ab). To divide monomials:
Steps:
Step 1: Divide the coefficients (numbers)
Step 2: For each variable, subtract the exponents (using the rule: xᵃ ÷ xᵇ = xᵃ⁻ᵇ)
Step 3: Combine the results
Example 1: 12x⁵ ÷ 3x²
Coefficients: 12 ÷ 3 = 4
x exponents: 5 – 2 = 3 → x³
Answer: 4x³
2. Dividing a Polynomial by a Monomial
A polynomial has two or more terms. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
Rule: (A + B + C) ÷ M = (A ÷ M) + (B ÷ M) + (C ÷ M)
Steps:
Step 1: Write the division as a sum of separate fractions
Step 2: Divide each term individually (using monomial division rules)
Step 3: Combine the results with the same signs
Example 1: (6x³ + 9x²) ÷ 3x
= (6x³ ÷ 3x) + (9x² ÷ 3x)
= (6÷3)x³⁻¹ + (9÷3)x²⁻¹
= 2x² + 3x
Answer: 2x² + 3x
3. Dividing a Polynomial by a Polynomial Using Factorization
When both dividend and divisor are polynomials, we can factor them first and then cancel common factors.
Steps:
Step 1: Factor the numerator (dividend) completely
Step 2: Factor the denominator (divisor) completely
Step 3: Cancel any common factors
Step 4: Write the simplified expression
Example 1: (x² + 5x + 6) ÷ (x + 2)
Factor numerator: x² + 5x + 6 = (x + 2)(x + 3)
So (x + 2)(x + 3) ÷ (x + 2) = x + 3
Answer: x + 3
4. Division with Remainder (When No Cancellation is Possible)
Sometimes the denominator does NOT factor into the numerator. In that case, we get a remainder. (For Grade 8, focus on cases where division is exact – no remainder.)
Example – No cancellation: (x² + 4x + 5) ÷ (x + 2)
The numerator does not factor with (x + 2) as a factor. This would leave a remainder (covered in polynomial long division in higher grades).
5. Cancellation of Common Factors
Before cancelling, remember:
- You can only cancel factors that are multiplied, not added
- Cancel the same factor from numerator and denominator completely
- Cancellation is valid only when the factor is not zero
Important: In (x² + 4x) ÷ x, you can factor x(x + 4) ÷ x = x + 4. But in (x² + 4) ÷ x, you cannot cancel because x is not a factor of the entire numerator.
Solved Examples
Example 1 – Monomial ÷ Monomial: Divide 24a⁵b³ by 6a²b
Solution: (24 ÷ 6) = 4; a: 5-2 = 3 → a³; b: 3-1 = 2 → b²
Answer: 4a³b²
Example 2 – Polynomial ÷ Monomial: Divide (9x³ + 6x² – 3x) by 3x
Solution: (9x³ ÷ 3x) + (6x² ÷ 3x) + (-3x ÷ 3x) = 3x² + 2x – 1
Answer: 3x² + 2x – 1
Example 3 – Using Factorization: Divide (x² + 7x + 12) by (x + 3)
Solution: Factor numerator: x² + 7x + 12 = (x + 3)(x + 4)
(x + 3)(x + 4) ÷ (x + 3) = x + 4
Answer: x + 4
Example 4 – Difference of Squares: Divide (9x² – 16) by (3x – 4)
Solution: 9x² – 16 = (3x – 4)(3x + 4)
(3x – 4)(3x + 4) ÷ (3x – 4) = 3x + 4
Answer: 3x + 4
Common Mistakes to Avoid
Mistake 1 – Subtracting exponents incorrectly
x⁵ ÷ x² = x³ (5 – 2 = 3), not x⁷ (which would be multiplication).
Correct understanding: When dividing, subtract exponents.
Mistake 2 – Dividing only the first term of a polynomial
(6x² + 9x) ÷ 3x = 2x + 3, not 2x² + 3x. You must divide EVERY term.
Correct understanding: Divide each term individually.
Mistake 3 – Cancelling terms that are not factors
(x² + 9x) ÷ x = x + 9 (correct), but (x² + 9) ÷ x cannot be simplified by cancelling x.
Correct understanding: You can only cancel factors that are multiplied, not added.
Mistake 4 – Forgetting to factor completely
(x² – 4) ÷ (x – 2) = (x – 2)(x + 2) ÷ (x – 2) = x + 2. Works if factored.
Correct understanding: Always factor the numerator before canceling.
Mistake 5 – Sign errors in factorization
x² – 5x + 6 = (x – 2)(x – 3), not (x + 2)(x + 3).
Correct understanding: Check signs when factoring quadratics.
Mistake 6 – Dividing by zero
If the divisor equals zero for some value, the division is undefined at that value.
Correct understanding: The simplified expression is valid only when the original divisor ≠ 0.
Quick Reference Summary
Monomial ÷ Monomial: Divide coefficients, subtract exponents
Polynomial ÷ Monomial: Divide each term of the polynomial by the monomial
Polynomial ÷ Polynomial: Factor both, then cancel common factors
Cancellation Rule: Cancel only factors (multiplied terms), not added terms
Key Formula: xᵃ ÷ xᵇ = xᵃ⁻ᵇ
Check: Division is exact when the divisor is a factor of the dividend